{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-07-15T00:25:42.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-07-15T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":61269,"title":"Precise Almost Pythagorean Triples ","description":"This  is essentially the same as:  Problem 52834. Easy Sequences 32: Almost Pythagorean Triples; it even presents the same set of test problems.  The difference is that the \"correct\" solutions for larger cases of 52834 were the result of roundoff errors due to values being larger than flintmax('double').  This problem requires care to avoid such roundoff.\r\nRepeating the original problem description:\t\t\t\t\r\nAn Almost Pythagorean Triple (abbreviated as \"APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is 1 less than the sum of square of the smaller elements (shorter sides). This means that if c is the hypotenuse and a and b are the shorter sides, , satisfies the following equation: \r\n        \r\n        where:  \r\nThe smallest  is the triple , with  and perimeter (the sum of the 3 elements)  of . Some researchers consider  as the smallest , but here, we will only look at 's where the hypotenuse is \"strictly\" greater than the other shorter sides. Other examples of 's are , and . \r\nUnfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible 's, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with 's with a known ratio between the hypotenuse and the shortest side: . \r\nGiven the value of r, find the perimeter of the  with the r-th smallest perimeter. For example for , that is , the smallest perimeter is  for  , while the second (r-th) smallest perimeter is , for the  with dimensions . For , the third smallest perimeter is  for  . \r\nThe output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.\r\nFinally, as with the original, the use of java, BigInteger, persistent, and global are not allowed.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 669.833px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 334px 334.917px; transform-origin: 334px 334.917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 42px; text-align: left; transform-origin: 310px 42px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 102.675px 7.91667px; transform-origin: 102.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis  is essentially the same as:  \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/52834\"\u003e\u003cspan style=\"border-block-end-color: rgb(0, 91, 130); border-block-start-color: rgb(0, 91, 130); border-bottom-color: rgb(0, 91, 130); border-inline-end-color: rgb(0, 91, 130); border-inline-start-color: rgb(0, 91, 130); border-left-color: rgb(0, 91, 130); border-right-color: rgb(0, 91, 130); border-top-color: rgb(0, 91, 130); caret-color: rgb(0, 91, 130); color: rgb(0, 91, 130); column-rule-color: rgb(0, 91, 130); outline-color: rgb(0, 91, 130); text-decoration-color: rgb(0, 91, 130); text-emphasis-color: rgb(0, 91, 130); \"\u003e\u003cspan style=\"\"\u003eProblem 52834. Easy Sequences 32: Almost Pythagorean Triples\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e; it even presents the same set of test problems.  The difference is that the \"correct\" solutions for larger cases of 52834 were the result of roundoff errors due to values being larger than flintmax('double').  This problem requires care to avoid such roundoff.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 10.5px; text-align: left; transform-origin: 310px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 201.933px 7.91667px; transform-origin: 201.933px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eRepeating the original problem description:\t\t\t\t\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 86.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 43.4583px; text-align: left; transform-origin: 310px 43.4583px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 303.05px 7.91667px; transform-origin: 303.05px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn Almost Pythagorean Triple (abbreviated as \"APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 7.91667px; transform-origin: 3.89167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.45px 7.91667px; transform-origin: 12.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eless\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.8917px 7.91667px; transform-origin: 94.8917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e than the sum of square of the smaller elements (shorter sides). This means that if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 7.91667px; transform-origin: 3.5px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ec\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.35px 7.91667px; transform-origin: 72.35px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the hypotenuse and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 7.91667px; transform-origin: 3.89167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ea\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 7.91667px; transform-origin: 3.89167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eb\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.3917px 7.91667px; transform-origin: 49.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are the shorter sides, \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 102.692px 7.91667px; transform-origin: 102.692px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, satisfies the following equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 24.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 12.4583px; text-align: left; transform-origin: 310px 12.4583px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5333px 7.91667px; transform-origin: 15.5333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"99\" height=\"19\" style=\"vertical-align: baseline;width: 99px;height: 19px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 23.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 11.9583px; text-align: left; transform-origin: 310px 11.9583px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.425px 7.91667px; transform-origin: 40.425px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        where:  \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"62\" height=\"18\" style=\"vertical-align: baseline;width: 62px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAAAkCAYAAABLw14kAAAC50lEQVR4nO2a7ZGDIBCG3x7swAZswAqsIB3YQTqwhdSQEuwhLViDLeR+wA4rB56woLnMPjPMmPOyfLzLskAARVEURVEURVGUT6AHcLNF+VJaAE8AK4C3LfdLW6QcYgDQCL5/hxO8L9IipQo3AAuMUJ3AzgQnuPKBcKHfMGG5FdgjW09504rQQNafryEktDQEt8zeKLQlpYFZXtYPaMul1BCaGJldybIggQtNbRkuasufdDCDz5OmHvKQRINQS2jiaW0v3t97lOnHHiGhpwp1kkbZdnlDn/Z5xjb5yZ0toUGoITRB9TzsZ79uEkGyC/BpA/WUFrqHGbfF1kXaJPelQzgzbuAav2Y08GyhYW3zEPoA8LLPg30uub63tg7fmUoK3cBFLX6mwHOVw4dLHZwgoRlMBh+Bd3uM2Ao945z9MI9IE35n6XyQZkE9ZwgNGLHJSX0HvSHRebnYIQ/hMzz1ePKsAfGhwVntcwhqT+z9Hmf3a0bcOSmCHo5UNDixjg9wncpd785KYoDt7H1H6uBOnLJHv8KB+QwW7za4sdh2gTqYMxN8zhCe92mK/A934pQ1fMB2d7Gg/jaL6pMsPb+M+VsXgs+W2ODlEBO+xH6ZEpvY7Aa2szQH//yglvAdCo7/EWN8YGokW6E9+QyZ8OREsVDNnVh6g1ZbeH54JLb7lzG+tZGs30fxBy9HeO7EsQRzLwHKpZbw/LZPbG/PGGXuFB5p/b6h/lmwP3gpovM+hcI5vV9Qx4F7OIcqITzPR2IO3O2828ATFx7Sexix+UHMBCdE7ZlOUH0pgvMDFb+dFNFegXelKSU8X35CSxRpdTjp5TPpZQuJDfZuxblic1IyeC44fa+DG/yzb6pCwqcuUzxq3eHuAuj0MEmTBmb2zjAe5IeG0b4bUw1fCB2lzrY8YPp1ZftJoFyno+/zPn3krRvd5kjKf3G0I7T48p9a8XCWW666y1YyoOVCUvQnQYqiKIqiKIqifAk/IsB4nRPGszsAAAAASUVORK5CYII=\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 96.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 48.3333px; text-align: left; transform-origin: 310px 48.3333px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.6167px 7.91667px; transform-origin: 41.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe smallest \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.725px 7.91667px; transform-origin: 37.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the triple \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"48\" height=\"18\" style=\"vertical-align: baseline;width: 48px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGEAAAAlCAYAAABSz4fZAAACsklEQVRoge2a0ZGDIBRFTw92YAM2kApSQTqwAztIC9awJdhDWrCGtJD9wDcSFQTDI+4OZ4afDAJ5Fy48FAqFQuH0dI5y+eag/gAX3LGL5gU8gGFRbilG+o+5sY7ZAxPPaF6UWZ+KC5lEqKb6W+UK3I8MwkPj6a8H6g/b9/2frVJ52somQjc94yopV1UFPD19DQn6uHnaX5YnftGziJAjKDY5BBcfDyl7/y+LCB0w4l+SqRDBW8U+JGh7h5B7YL0sIjxJ7/kuWvaX/6f0hIk8Bo5FXQTxzhazWWozYqxibzP8hJBjeEO41aqLMLLepHp0BNnaLB/T7zms0EasKCQJUxVh7wTRkzY4S8GX4ufMa2QsIbaovhLqqV4L/LB9ckgphJ13bIlyTdiXC7GiR2D9bHmCULMW49C9SSA33o/HT/StKcaK4AsiCLZVaQem4v1cryk6xFkRfFEEmGdMjnuomnlFpE4ObWKtCL4sQs0sgmZyJfTMK0+LWCuCL4sAs03kEOGKvgixVgQnEGEgLLVPgfzZUbn9GCuyn4sm9UrQvGYQZCX0Su2L3cVu/KfYE2JnzlEkSFq5gmz8sRNKVYRmp44EZatOhdknOsKuOOQliysAIrjrZCTvfo9eccgqO2J1aiK0zCefkXUgJU9wLV0RSMqeEHZC1vEeSMkTHo527FPaC5NMxiLjPXJbrCaCNLy8K+ows3HEbwvLrHrPxwfe649TX3fmS0PXDG9YjzXWUmQSHLFoVTuqMUGwvyyQJb9HgxFiYA7iHjfrmYFZ9JCAyuQ4clBopmePrCA4wekohJyf0lzQO8L6+jy1CA35Xo2Cmc05EkebU4vQ4N5MNejJ9xrW5iMRWtbf16RKuipMQHIkcWDsLoflyfsVu8hJMhrXGyzta+K/ju9znEKhUCgUCmfnFzTbYsvcMy9LAAAAAElFTkSuQmCC\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.275px 7.91667px; transform-origin: 18.275px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, with \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"100\" height=\"19\" style=\"vertical-align: baseline;width: 100px;height: 19px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 101.508px 7.91667px; transform-origin: 101.508px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and perimeter (the sum of the 3 elements)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 7.91667px; transform-origin: 7.775px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eof \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"18\" height=\"18\" style=\"vertical-align: baseline;width: 18px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA6UlEQVRYhe2WQQ3DMAxFP4cwCIESGIIiKIMxKINSKIZBKIdRKIZR6A6ztWiHxkm+qkjzl6xenOQpL5UMeDyevhIBhEzPAOBmrKEFZAVwGDbZpc9SSylISEC0zoDGApgDn1syZwQwy/dpBNqk4klPkH32EpjfzAagKIfk3tiESl01QBYFD1ToqgGyRHW9kL/JS4BU19oCwwRSXWMPQDRdLCCaLhYQTRcDiKqLAUTVxQDaQNTVChRB1tUKdAdZVyuQTgpTD0CprrORpCg6VijQXLBWdW0skAXfoSutVQ7LRdfT/i6Px+P5u7wB3c96XQFb71kAAAAASUVORK5CYII=\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91.4083px 7.91667px; transform-origin: 91.4083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Some researchers consider \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"48\" height=\"18\" style=\"vertical-align: baseline;width: 48px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.5583px 7.91667px; transform-origin: 50.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as the smallest \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.7917px 7.91667px; transform-origin: 70.7917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, but here, we will only look at \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 243.942px 7.91667px; transform-origin: 243.942px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e's where the hypotenuse is \"strictly\" greater than the other shorter sides. Other examples of \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.8333px 7.91667px; transform-origin: 18.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e's are \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"56\" height=\"18\" style=\"vertical-align: baseline;width: 56px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 7.91667px; transform-origin: 17.5px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"71\" height=\"18\" style=\"vertical-align: baseline;width: 71px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 71.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 35.875px; text-align: left; transform-origin: 310px 35.875px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 269.142px 7.91667px; transform-origin: 269.142px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eUnfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.9417px 7.91667px; transform-origin: 21.9417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e's, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.775px 7.91667px; transform-origin: 6.775px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e's with a known ratio between the hypotenuse and the shortest side: \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"50\" height=\"18\" style=\"vertical-align: baseline;width: 50px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 95.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 47.8333px; text-align: left; transform-origin: 310px 47.8333px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.2083px 7.91667px; transform-origin: 62.2083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven the value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2.33333px 7.91667px; transform-origin: 2.33333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003er\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.925px 7.91667px; transform-origin: 85.925px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, find the perimeter of the \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3167px 7.91667px; transform-origin: 30.3167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e with the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.6167px 7.91667px; transform-origin: 41.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; text-decoration-line: underline; \"\u003er-th smallest\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.3417px 7.91667px; transform-origin: 37.3417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e perimeter. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.4417px 7.91667px; transform-origin: 12.4417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example for \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"35\" height=\"18\" style=\"vertical-align: baseline;width: 35px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.4917px 7.91667px; transform-origin: 24.4917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, that is \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"43\" height=\"18\" style=\"vertical-align: baseline;width: 43px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.675px 7.91667px; transform-origin: 81.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the smallest perimeter is \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"18\" height=\"18\" style=\"vertical-align: baseline;width: 18px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAABLUlEQVRYhe2WUbHDIBBFjwccYKAGoiAK4qAOcBAL0RAJeKiFaoiFvg+WmX2dPEIC5fWDO7Nfu1lOLiwJdHV1fZcsYE4+M0jcaoMswCuzsQFm4Ak4iafEvQTEKJAYR0AWeMji726u0sPv5A41Et5slAVygWKt28kZYEvks+UygaaMOu34aZfOAsUt2RI1GvqyS7lAcTtSQIPqtX4aKNakgGxmXVWgo/PRDMiruvEbgO4cn4+mW2YIF2LKJT32j08DIXkN5eX5meDIqnJLCyAITk2yoBcIR9iumbxzVhUoBRrvqr1vXXMg3Wcq6FMF6MbxBDYD0tN36dejJpCeurUGjOX3GJ/5QjvCId4o/FOMIDPB4vdY/ljAEEY53jle6opduSorAMN/QnR1dXWV6Ae9sZqCc2SAGwAAAABJRU5ErkJggg==\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.05px 7.91667px; transform-origin: 12.05px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"48\" height=\"18\" style=\"vertical-align: baseline;width: 48px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.9583px 7.91667px; transform-origin: 57.9583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, while the second (r-th) smallest perimeter is \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"18\" height=\"18\" style=\"vertical-align: baseline;width: 18px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA6klEQVRYhe2WUQ2DMBCGfw91UAMYmIIqwMEc4AAL1TAJ9TALaJgF9sBd0i2DXQ9u4+G+5MIDBT74QgBwHOc/dAAuwumE54wAglZoAjALZxSIZForlX8hNcjMWJ7SJ0IlwqMSKjRxY02gC0wr+xOAgbb3PUKRLvKtdQ9ZLpDYLqG1BDU3bOc6TEgC53pA9taYC3GuLFxvLsS50hmEWnOZC7XmMhdqzWUqpMllKqTJZSpU0J7LTChCl8tM6ApdLjMh/mL3ZxCqc239kvxMiHMVxbH8O8NCwxFCI8m0vF2xOu59MpabdBzHcSQ8AUZlel3qgGXQAAAAAElFTkSuQmCC\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.6583px 7.91667px; transform-origin: 25.6583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, for the \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.6833px 7.91667px; transform-origin: 53.6833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with dimensions \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"71\" height=\"18\" style=\"vertical-align: baseline;width: 71px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.325px 7.91667px; transform-origin: 16.325px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"35\" height=\"18\" style=\"vertical-align: baseline;width: 35px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAAkCAYAAADb0CfrAAABd0lEQVRoge2YX5GDMBCHPw84wAAGogAFdYADHNQCGiohHrBwGmqBewg7hOtw14WUpL39ZvahD8ns/rr/CBiGYRiGURY14IAmtyMl0QNfwBTZHehyOlUCA0GMgSDSQBBGROrzuZaXHvA8llEFjCwZVJ3sV3ZEgK3ALyzZ8+96UENovls4lsypT/HojZDMuR29yLFW969/5R3whAmmLqkuOix1Wc3m2dfpXQJL1RsGQmyHykk6ujS2kbVgV8VdUwI7WgJtFNM0x7K7AuSi6+yYiCG/Ncr7BLZ3J6kJ2eJZ7zhirfbCinWGjDsdKxHHYwapdp32x+FP3AVigVTlJWt3knFXKHECqL6z4sabImtKmlYxanEa0mdNCdPqN7+eFr6LDqk7+QY5p9UWNcuq8jSy6N0TO1MaPcoEiEf48CKnzkCa7dajlnxbaRbZVQdPVVI5iJ8kZB3pZ7sRSkkdn3xX+WRu5uNCEELikdfAT9zZDMMwDMMwjBR8AxS7vP5/0oaBAAAAAElFTkSuQmCC\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.1083px 7.91667px; transform-origin: 31.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the third smallest perimeter is \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"33\" height=\"18\" style=\"vertical-align: baseline;width: 33px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.05px 7.91667px; transform-origin: 12.05px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"101\" height=\"18\" style=\"vertical-align: baseline;width: 101px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 21px; text-align: left; transform-origin: 310px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 305.717px 7.91667px; transform-origin: 305.717px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 21px; text-align: left; transform-origin: 310px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 282.908px 7.91667px; transform-origin: 282.908px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFinally, as with the original, the use of java, BigInteger, persistent, and global are not allowed.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function function perimeter = rthPerAPTdbl(r)\r\n  perimeter = r^2;\r\nend","test_suite":"%% Test Case 1\r\nr = 2;\r\np_correct = 71;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 2\r\nr = 3;\r\np_correct = 1393;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 3\r\nr = 5;\r\np_correct = 1046629;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 4\r\nr = 10;\r\np_correct = 737287485879;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 5\r\nr = 100;\r\np_correct = 16183149010201;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 6\r\nrs = 101:150;\r\nps = arrayfun(@(r) rthPerAPTdbl(r),rs);\r\nps = mod([sum(ps) ps(5:5:end) floor(std(double(ps)))],1e6);\r\nps_correct = [12636 824229 203679 227761 926641 15749 664839 210241 515881 139269 477199 789840];\r\nassert(isequal(ps,ps_correct))\r\n%% Test Case 7\r\nr = 1000;\r\np_correct = 499499001002001;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 8\r\nr = 10000;\r\np_correct = 100020001;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 9\r\nr = 123456;\r\np_correct = uint64(76696064606196865);\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 10: Forbid java and BigInteger\r\nfiletext = fileread('rthPerAPTdbl.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'BigInteger') || contains(filetext, 'java'); \r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":5,"created_by":2404920,"edited_by":2404920,"edited_at":"2026-03-04T15:24:47.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2026-03-04T14:14:29.000Z","updated_at":"2026-07-19T13:59:15.000Z","published_at":"2026-03-04T15:24:47.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis  is essentially the same as:  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/52834\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 52834. Easy Sequences 32: Almost Pythagorean Triples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e; it even presents the same set of test problems.  The difference is that the \\\"correct\\\" solutions for larger cases of 52834 were the result of roundoff errors due to values being larger than flintmax('double').  This problem requires care to avoid such roundoff.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eRepeating the original problem description:\\t\\t\\t\\t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn Almost Pythagorean Triple (abbreviated as \\\"APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eless\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e than the sum of square of the smaller elements (shorter sides). This means that if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the hypotenuse and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are the shorter sides, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, satisfies the following equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"19\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"99\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        where:  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"62\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe smallest \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the triple \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"48\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"19\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"100\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and perimeter (the sum of the 3 elements)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eof \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId6\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Some researchers consider \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"48\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId7\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the smallest \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, but here, we will only look at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e's where the hypotenuse is \\\"strictly\\\" greater than the other shorter sides. Other examples of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e's are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"56\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId8\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"71\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId9\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUnfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e's, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e's with a known ratio between the hypotenuse and the shortest side: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"50\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId10\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven the value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, find the perimeter of the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e with the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er-th smallest\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e perimeter. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"35\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId11\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, that is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"43\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId12\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the smallest perimeter is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId13\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"48\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId14\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, while the second (r-th) smallest perimeter is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId15\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, for the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with dimensions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"71\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId16\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"35\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId17\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the third smallest perimeter is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"33\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId18\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"101\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId19\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFinally, as with the original, the use of java, BigInteger, persistent, and global are not 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sum","description":"Given natural number calculate its _population count_.","description_html":"\u003cp\u003eGiven natural number calculate its \u003ci\u003epopulation count\u003c/i\u003e.\u003c/p\u003e","function_template":"function population = popcount(n)\r\n  binary = dec2bin(n);\r\n  population = ???\r\nend","test_suite":"%%\r\nN = 1; \r\nnum_ones = 1;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 2; \r\nnum_ones = 1;\r\nassert(isequal(popcount(N),num_ones))\r\n%%\r\nN = 3; \r\nnum_ones = 2;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nfor num_ones = 3:10\r\n  N = 2^num_ones-1;\r\n  assert(isequal(popcount(N),num_ones))\r\nend\r\n\r\n%%\r\n\r\nfor num_ones = 1:50\r\n  N = sum(2.^(find(randperm(50)\u003c=num_ones)-1));\r\n  assert(isequal(popcount(N),num_ones))\r\nend\r\n\r\n%%\r\nN = 23; \r\nnum_ones = 4;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 123; \r\nnum_ones = 6;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 123456; \r\nnum_ones = 6;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 1234567; \r\nnum_ones = 11;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 1125899906842501;\r\nnum_ones = 45;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":76,"test_suite_updated_at":"2015-01-19T23:10:17.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2014-12-08T11:59:42.000Z","updated_at":"2026-03-09T20:12:46.000Z","published_at":"2015-01-19T11:07:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven natural number calculate its\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epopulation count\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46000,"title":"Compute the harmonic numbers","description":"The nth \u003chttps://mathworld.wolfram.com/HarmonicNumber.html harmonic number\u003e is defined as the sum of the reciprocals of the integers from 1 to n: \r\n\r\n\u003c\u003chttps://mathworld.wolfram.com/images/equations/HarmonicNumber/NumberedEquation1.gif\u003e\u003e\r\n\r\nThe first four harmonic numbers are 1, 3/2, 11/6, and 25/12. The harmonic numbers appear in several interesting identities, including this one for \u003chttps://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant Apery's constant\u003e:\r\n\r\n\u003c\u003chttps://mathworld.wolfram.com/images/equations/RiemannZetaFunction/NumberedEquation34.gif\u003e\u003e\r\n\r\nCompute the nth harmonic number.\r\n\r\n\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 210px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 105px; transform-origin: 407px 105px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 7.8px; transform-origin: 14px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 7.8px; transform-origin: 5.83333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/HarmonicNumber.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eharmonic number\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 201.083px 7.8px; transform-origin: 201.083px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined as the sum of the reciprocals of the integers from 1 to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Definition of the harmonic numbers: sum(1/k) from k = 1 to k = n\" style=\"width: 69px; height: 45px;\" width=\"69\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 381.85px 7.8px; transform-origin: 381.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe first four harmonic numbers are 1, 3/2, 11/6, and 25/12. The harmonic numbers appear in several interesting identities, including this one for\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eApery's constant\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Relation between zeta(3) and the harmonic numbers\" style=\"width: 98.5px; height: 45px;\" width=\"98.5\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 42.0167px 7.8px; transform-origin: 42.0167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCompute the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.8px 7.8px; transform-origin: 63.8px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth harmonic number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function H = HarmonicNumber(n)\r\n  H = f(n);\r\nend","test_suite":"%%\r\nn = 1;\r\nH_correct = 1;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 2;\r\nH_correct = 3/2;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 3;\r\nH_correct = 11/6;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 4;\r\nH_correct = 25/12;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 5;\r\nH_correct = 137/60;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 10;\r\nH_correct = 7381/2520;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 15;\r\nH_correct = 1195757/360360;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 20;\r\nH_correct = 55835135/15519504;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 29;\r\nH_correct = 9227046511387/2329089562800;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nfiletext = fileread('HarmonicNumber.m');\r\nx = ~isempty(strfind(filetext, 'A002805')) || ~isempty(strfind(filetext, 'A001008')) ||...\r\n    ~isempty(strfind(filetext, '360360'));\r\nassert(~x, 'Illegal approach')","published":true,"deleted":false,"likes_count":4,"comments_count":1,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":96,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-29T03:02:15.000Z","updated_at":"2026-07-09T05:57:38.000Z","published_at":"2020-06-29T03:50:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/HarmonicNumber.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eharmonic number\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is defined as the sum of the reciprocals of the integers from 1 to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Definition of the harmonic numbers: sum(1/k) from k = 1 to k = n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eH_n = \\\\sum_{k = 1}^n\\\\frac{1}{k}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first four harmonic numbers are 1, 3/2, 11/6, and 25/12. The harmonic numbers appear in several interesting identities, including this one for\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eApery's constant\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Relation between zeta(3) and the harmonic numbers\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\zeta(3) = \\\\frac{1}{2}\\\\sum_{k = 1}^\\\\infty\\\\frac{H_k}{k^2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth harmonic number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":43082,"title":"Reverse digits of a number","description":"Reverse digits of a number.\r\n\r\nFor example, return *321* for the input *123* .\r\n\r\nThe left-hand zeros can be discarded. For example, input *10* will return *1* .","description_html":"\u003cp\u003eReverse digits of a number.\u003c/p\u003e\u003cp\u003eFor example, return \u003cb\u003e321\u003c/b\u003e for the input \u003cb\u003e123\u003c/b\u003e .\u003c/p\u003e\u003cp\u003eThe left-hand zeros can be discarded. For example, input \u003cb\u003e10\u003c/b\u003e will return \u003cb\u003e1\u003c/b\u003e .\u003c/p\u003e","function_template":"function y = reverseDigits(x)\r\n    y = x;\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = 1;\r\nassert(isequal(reverseDigits(x),y_correct))\r\n\r\n%%\r\nx = 10;\r\ny_correct = 1;\r\nassert(isequal(reverseDigits(x),y_correct))\r\n\r\n%%\r\nx = 675;\r\ny_correct = 576;\r\nassert(isequal(reverseDigits(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":25354,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":77,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-10-05T22:52:12.000Z","updated_at":"2026-05-28T16:01:59.000Z","published_at":"2016-10-05T22:52:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReverse digits of a number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e321\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for the input\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e123\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e .\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe left-hand zeros can be discarded. For example, input\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e .\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2369,"title":"Tribute to Ramanujan","description":"The nth taxicab number, denoted as T(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways (source: wikipedia).\r\n\r\nExample: \r\n\r\n           T(1) = 2 = 1^3 + 1^3\r\n           T(2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3\r\n\r\nReturn the value of n if the given number is a taxicab number. Return 0 if not. If the input is 1729, output would be 2, since 1729 is smallest number that can be expressed as a sum of two cubes in two (n=2) distinct ways.\r\n\r\nAvoid look up table solution. Test suit might expand.\r\n\r\n","description_html":"\u003cp\u003eThe nth taxicab number, denoted as T(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways (source: wikipedia).\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e           T(1) = 2 = 1^3 + 1^3\r\n           T(2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3\u003c/pre\u003e\u003cp\u003eReturn the value of n if the given number is a taxicab number. Return 0 if not. If the input is 1729, output would be 2, since 1729 is smallest number that can be expressed as a sum of two cubes in two (n=2) distinct ways.\u003c/p\u003e\u003cp\u003eAvoid look up table solution. Test suit might expand.\u003c/p\u003e","function_template":"function y = mytaxicab(x)\r\n\r\n\r\n\r\nend","test_suite":"%%\r\nx = 2;\r\ny_correct = 1;\r\nassert(isequal(mytaxicab(x),y_correct))\r\n\r\n%%\r\nx = 1729;\r\ny_correct = 2;\r\nassert(isequal(mytaxicab(x),y_correct))\r\n\r\n%%\r\nx = 5251;\r\ny_correct = 0;\r\nassert(isequal(mytaxicab(x),y_correct))\r\n\r\n%%\r\nx = 10000001;\r\ny_correct = 0;\r\nassert(isequal(mytaxicab(x),y_correct))\r\n\r\n\r\n%%\r\n\r\nx = 87539319;\r\ny_correct = 3;\r\nassert(isequal(mytaxicab(x),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":6,"created_by":17203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":33,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2014-06-18T06:26:28.000Z","updated_at":"2026-05-28T02:36:30.000Z","published_at":"2014-06-18T06:27:24.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe nth taxicab number, denoted as T(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways (source: wikipedia).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[           T(1) = 2 = 1^3 + 1^3\\n           T(2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn the value of n if the given number is a taxicab number. Return 0 if not. If the input is 1729, output would be 2, since 1729 is smallest number that can be expressed as a sum of two cubes in two (n=2) distinct ways.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAvoid look up table solution. Test suit might expand.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46305,"title":"find the value","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"display: block; min-width: 0px; padding-top: 0px; vertical-align: baseline; perspective-origin: 407px 10.5px; transform-origin: 407px 10.5px; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; perspective-origin: 384px 10.5px; transform-origin: 384px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003epi*(2+x)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = your_fcn_name(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 5;\r\ny_correct = 7*pi;\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":424112,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":116,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-08-14T07:42:35.000Z","updated_at":"2026-05-30T15:28:23.000Z","published_at":"2020-08-14T07:42:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003epi*(2+x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":43130,"title":"Nth root","description":"Nth root of a number x","description_html":"\u003cp\u003eNth root of a number x\u003c/p\u003e","function_template":"function y = nroot(n,x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 1;\r\nn=1;\r\ny_correct = 1;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n%%\r\nx = 1;\r\nn=2;\r\ny_correct = 1;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n\r\n%%\r\nx = 1;\r\nn=2;\r\ny_correct = 1;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n\r\n%%\r\nx = 10;\r\nn=2;\r\ny_correct = 3.1623;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n%%\r\nx = 10;\r\nn=10;\r\ny_correct = 1.2589;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n%%\r\nx = 100;\r\nn=100;\r\ny_correct = 1.0471;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":91311,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":138,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-10-06T17:19:12.000Z","updated_at":"2026-02-17T09:14:29.000Z","published_at":"2016-10-06T17:19:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNth root of a number x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":42865,"title":"Find the sum of n squares","description":"What is the sum of the squares of the first n integers?","description_html":"\u003cp\u003eWhat is the sum of the squares of the first n integers?\u003c/p\u003e","function_template":"function y = your_fcn_name(x)\r\n  y = x^2;\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = 1;\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n\r\nx = 15;\r\ny_correct = 1240;\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n\r\nx = 50;\r\ny_correct = 42925;\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":69320,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":210,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-06-02T15:33:45.000Z","updated_at":"2026-07-07T03:36:09.000Z","published_at":"2016-06-02T15:33:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the sum of the squares of the first n integers?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":54370,"title":"Determine whether the given number is palindrome or not.","description":"A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed. \r\nFor more: https://en.wikipedia.org/wiki/Palindromic_number#:~:text=A%20palindromic%20number%20(also%20known,when%20its%20digits%20are%20reversed.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003epalindromic number\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (also known as a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enumeral palindrome\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enumeric palindrome\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e) is a number (such as 16461) that remains the same when its digits are reversed. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor more: https://en.wikipedia.org/wiki/Palindromic_number#:~:text=A%20palindromic%20number%20(also%20known,when%20its%20digits%20are%20reversed.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = palindrome(x)\r\ny=x\r\nend","test_suite":"%%\r\nx = 16465;\r\ny_correct = false;\r\nassert(isequal(palindrome(x),y_correct))\r\n%%\r\nx = 16461;\r\ny_correct = true;\r\nassert(isequal(palindrome(x),y_correct))\r\n%%%\r\nx = 12321;\r\ny_correct = true;\r\nassert(isequal(palindrome(x),y_correct))\r\n%%%%%\r\nx = 54353453;\r\ny_correct = false;\r\nassert(isequal(palindrome(x),y_correct))\r\n%%%%%\r\nx = 88888888888888;\r\ny_correct = true;\r\nassert(isequal(palindrome(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":1780615,"edited_by":1780615,"edited_at":"2022-04-27T16:41:00.000Z","deleted_by":null,"deleted_at":null,"solvers_count":34,"test_suite_updated_at":"2022-04-27T16:41:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-04-27T16:18:37.000Z","updated_at":"2026-06-01T04:34:54.000Z","published_at":"2022-04-27T16:41:00.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epalindromic number\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (also known as a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumeral palindrome\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e or a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumeric palindrome\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e) is a number (such as 16461) that remains the same when its digits are reversed. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor more: https://en.wikipedia.org/wiki/Palindromic_number#:~:text=A%20palindromic%20number%20(also%20known,when%20its%20digits%20are%20reversed.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46639,"title":"Transform a Decimal number into a Septimal number ","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 211.4px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 105.7px; transform-origin: 407px 105.7px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 62.4px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.2px; text-align: left; transform-origin: 384px 31.2px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSome scientists believes that a septimal system (7 base number system) was in use approximately 2500 years ago by the Magyar (Hungarian) tribes. Though some are disagreed. However, How will it be if you try to convert some numbers to their number sytem (Septimal Number System) from ours (Decimal Number System). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eLet me it crystal clear by incluing an example.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eLet,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eDecimal number, x = 93 (as Input)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSeptical Number, y = 162 (as Output)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e which is equivalent to the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eDecimal number, x = 93 (Input)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAs you have to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003econvert a decimal number into a septimal number\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e ,you must show the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eoutput as y\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (a septimal number).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y=DecimaltoSeptimal(x)\r\ny=[];\r\nend\r\n","test_suite":"%%\r\nx =7;\r\ny= 10;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =93;\r\ny= 162;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =22;\r\ny= 31;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =20145;\r\ny= 112506;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =2014;\r\ny= 5605;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =107033;\r\ny= 624023;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":430818,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-10-01T19:42:08.000Z","updated_at":"2026-05-30T17:40:40.000Z","published_at":"2020-10-01T19:49:16.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSome scientists believes that a septimal system (7 base number system) was in use approximately 2500 years ago by the Magyar (Hungarian) tribes. Though some are disagreed. However, How will it be if you try to convert some numbers to their number sytem (Septimal Number System) from ours (Decimal Number System). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet me it crystal clear by incluing an example.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDecimal number, x = 93 (as Input)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSeptical Number, y = 162 (as Output)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e which is equivalent to the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDecimal number, x = 93 (Input)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs you have to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003econvert a decimal number into a septimal number\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ,you must show the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eoutput as y\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (a septimal number).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":2370,"title":"Fermat's last theorem - again","description":"For a given integer n, express it as a sum of two squares if possible. Return empty matrix otherwise. Solution may not be unique and uniqueness will not be checked, so return one of the solutions.\r\nExample: \r\n\r\n  n=25\r\n  Output=[3 4]\r\n\r\n\r\n  n=41\r\n  Output=[5 4]\r\n \r\n  n=7\r\n  Output=[]","description_html":"\u003cp\u003eFor a given integer n, express it as a sum of two squares if possible. Return empty matrix otherwise. Solution may not be unique and uniqueness will not be checked, so return one of the solutions.\r\nExample:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003en=25\r\nOutput=[3 4]\r\n\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003en=41\r\nOutput=[5 4]\r\n\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003en=7\r\nOutput=[]\r\n\u003c/pre\u003e","function_template":"function y = sumOfSqr(x)\r\n\r\n\r\nend","test_suite":"%%\r\nx = 25;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))\r\n\r\n\r\n%%\r\nx = 41;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))\r\n\r\n%%\r\nx = 7;\r\ny=sumOfSqr(x);\r\nassert(isempty(y));\r\n\r\n%%\r\nx = 391250;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))\r\n\r\n%%\r\nx = 288;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))\r\n\r\n%%\r\nx = 1;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":17203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":42,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2014-06-18T06:53:27.000Z","updated_at":"2026-05-28T02:36:32.000Z","published_at":"2014-06-18T06:54:02.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer n, express it as a sum of two squares if possible. Return empty matrix otherwise. Solution may not be unique and uniqueness will not be checked, so return one of the solutions. Example:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[n=25\\nOutput=[3 4]\\n\\nn=41\\nOutput=[5 4]\\n\\nn=7\\nOutput=[]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":45189,"title":"Double Factorial","description":"Return double factorial\r\n\r\n        n(n-2)...(5)(3)(1),   n\u003e0, odd\r\nn!!  =  n(n-2)...(6)(4)(2),   n\u003e0, even\r\n        1                 ,   n = -1, or n = 0\r\n\r\nFor example\r\n\r\n6!! = (6)(4)(2) = 48","description_html":"\u003cp\u003eReturn double factorial\u003c/p\u003e\u003cpre\u003e        n(n-2)...(5)(3)(1),   n\u0026gt;0, odd\r\nn!!  =  n(n-2)...(6)(4)(2),   n\u0026gt;0, even\r\n        1                 ,   n = -1, or n = 0\u003c/pre\u003e\u003cp\u003eFor example\u003c/p\u003e\u003cp\u003e6!! = (6)(4)(2) = 48\u003c/p\u003e","function_template":"function y = doubleFactorial(x)\r\n  y = x;\r\nend","test_suite":"\r\ncorrectList = [1 1 2 3 8 15 48 105 384 945 3840 10395 46080 135135 645120 2027025 10321920];\r\nfor idx = 1:17\r\nassert(isequal(doubleFactorial(idx-1),correctList(idx)))\r\nend","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":8703,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":69,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2019-11-01T18:08:07.000Z","updated_at":"2026-05-30T02:33:35.000Z","published_at":"2019-11-01T18:08:34.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn double factorial\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[        n(n-2)...(5)(3)(1),   n\u003e0, odd\\nn!!  =  n(n-2)...(6)(4)(2),   n\u003e0, even\\n        1                 ,   n = -1, or n = 0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e6!! = (6)(4)(2) = 48\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60948,"title":"Check p² = 24k + 1, p \u003e 3, the 'golden prime squares' equation","description":"Historical context\r\nIn december 2023, I / Nicolas Douillet was working on prime squares properties and I found* the formula :\r\n\r\n                                                   \r\n\r\nwhich I named the ‘Christmas eve equation'. From it I then wrote :\r\n\r\n\r\n\r\nand I named it the ‘golden prime squares equation’. These two formulas, which are actually the same one, can easily be proven starting from  (note this last formula applies not only to prime numbers). \r\n \r\n* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website :  Site de Gérard Villemin\r\n \r\nProblem statement\r\nYour goal here is to check this formula for the first primes less or equal to a given integer m (input). To do so you will simply compute k in a vector. For the two first cases where the formula doesn’t apply -2 and 3- your function should return 0.\r\n \r\nExamples\r\n\r\nIf m = 20 then k = [0, 0, 1, 2, 5, 7, 12, 15], since 5^2 = 24x1 + 1, 7^2 = 24x2 + 1, 11^2 = 24x5 + 1, etc…\r\n\r\nIf m = 40 then k = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ; \r\n \r\nFobidden functions\r\n \r\nregexp\r\nstr2num\r\nassignin\r\n \r\nSee also (my file exchange contributions using this equation) :\r\n\r\nIsprime optimus [FAST]\r\nPrime number spiral (my corresponding contribution on my file exchange)\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1019.47px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 509.733px; transform-origin: 408px 509.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0583px 8px; transform-origin: 68.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn december 2023, I / \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/profile/authors/8668631?s_tid=user_nav_comm\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eNicolas Douillet\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 210.05px 8px; transform-origin: 210.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e was working on prime squares properties and I found* the formula :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"251\" height=\"19\" style=\"width: 251px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.025px 8px; transform-origin: 99.025px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                                   \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.7333px 8px; transform-origin: 58.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhich I named the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.4167px 8px; transform-origin: 76.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e‘Christmas eve equation'\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0417px 8px; transform-origin: 68.0417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. From it I then wrote :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"249\" height=\"19\" style=\"width: 249px; height: 19px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.95px 8px; transform-origin: 57.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand I named it the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.2167px 8px; transform-origin: 99.2167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e‘golden prime squares equation’\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.483px 8px; transform-origin: 215.483px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. These two formulas, which are actually the same one, can easily be proven starting from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN8AAAAkCAYAAAAJrO/2AAAHOklEQVR4Xu2cS8iuUxTHvzMnYURCGFBu5VpiIFFiQq4DnSKXgYEBoZNOJyGSZOASOsnAJUpKuYycjlxSLokBhcTILebO+mWvWmd/ez/P3s+7n/t+avW+3/s+l73/e/3Xdb/fjq16VAQqAqMgsGOUp9aHVgQqAluVfFUJKgIjIVDJNxLw9bEVgUq+qgMVgZEQqOQbCfj62IpAJd92HThHPjpB5CyRP0UermpSEdgQgdvl+vdFfrD3qeT7H40TRe4RuVLkb5G3RV4X+WxD0Ody+aUy0D0ib1Vj03nJfpUrj4pc/Z18for/XSXf1tajAsrdIv+KPC3yiMgfnZdgXhcq6c5zw75/xuTDgB7kWQZcCjwbuhM7griumXxHCFLviKB4v4ncJfLqgAs25qOY+70ix4mcLnLyAsj3sczh2JHWEa+3T+SFyKK+F/p8zeT71ikdHu+WFREPPYB8HHh4vN+7CyAfhvNaN48hjSle7zGRi0Wy0pS1ku9ZAerWBShcCe+5FPKBhR9GD0FCPO5hIttyurbFWSP5rLKxOKc6D9CG1VjfYyjOEHmiJ++8JPLpGoVISEHpmcKLaLHj1hRWPhShjtCaf+aSj6SWkEXdK+/PdhMKxrWFJ1vidoRYgMZBuECVc8oHlpW8tK9iyFjko6VD2KtKim4htHeywreGxfNJqEW1UmuuaxMawify4eVujsEhtpEPMG4ToQSvSTkku0GE6o7G2HrzO+RNaetSkhjM53tzw+vl/VUiF4pQJmZxPhJ5UcQvvqihof93pgjFCo7z3et18rrb4cR9snOAyESXQj6IcLXTJS3JY1A+EHlKRCuuwFASP+7XFwm57yUihJ0XubW3y8g80JWgF2wjn94IxXrF/fGavKKsuHHeHy7yhcgh7vtz5TXVcv0XUbicjzEGlyVe4JeECRO+EvnJAXijmcdz8h7DoweW+hoRa4jUG0FUMKE/qLkklk+JmTi84GlLIZ9OzubbYIzS7hYBQ6tnKO6hmwAXuLYvEuqjMO4PiVinRGpzdGgeqeSzSsvNdorYMLNrAWNo8mlPDywg3hUi1irh3faLNJXe7T1OcmBzP7w+YZTOKccoNOnY0sinVUnI9UtgDbQKDSY5hjyHp32TkPu/IaIOKRgRppLPlnEJ1fyQzCqk7zFyQOn7XBujx0JkG5qGrJbeA8+21xkije3ttaVC8KWRT3eCxEIyu0YhXSuhI76HYiz0fNWAlniGzaWDhjiVfP84FuMtLnAW3g7Qkq+vwkAJQOzCNlnVJutrPdtpMijCTfWeNmzCK7ZWvOQci90mc+zqJYYsuBC6f+omScoCXv6RukZdsPJJh3GlOY7OpqxV7jO1uBcMoVPIZwGLVQdtBZH8a6qVz9SFjYXRllwshG+ZNUII7uWLrNyayHefYEBOxBHTEzX0KCxFrRJb/UKke1nu3fdWQk3XOpPPAhazrgrY1PtmVtGbPEXMk1tShkIJDalKtjCWFHaqkY7piTX0JXLmsUindlajis7k08WPhZy2GJMbcg5dcEkdqyWf9W5N+YrN90p6/yWRT9c7FnLa2kLXMBrFH5t0Pvk653wKWKyQ0kbOpjh5aPJRzfzdDaipMKQezvabLLlCYGqEUNr7L4V8NrcMFVJYG1o+VAhj5GzLuXzS4TBeEmE9S4Svbc/3v1c96lTtbANMFa50UzR3kjnn21J3rI9kK5raq2vLVzSkUmKCHT3QTX8pMTXyMa8uu1BsNHFkgAyKXyzCaltjv430pFww5oYPNSakZMEtjG0FF5vj+IBpfA7x2Lkw1SKLv2g2rwiFyerh/Hk15SvWo2LlPncW1+8jtilQ6Pspkc8WrHI3EWgFOVSMUsMW6r2mYqYbnIcinWIR6nszZowuLajoL2bayGdL7pZ8VP2eF/lG5E6R1B0tqUD2fZ7N/WwIpE32Y2QA/MDWWk4tKoVCIr+nQwtiZyGD1Df5cnq0fpqQ2oezxsknnz6fNCBpQ3JEOdDJTaOMHL3T/F+vwfk8LkK0w29D2XJ2UxM3mshnAYPdX4t8KcIO+79E3hx4sjnApJyr+QF9Oo6f3St5h9/3seSKNc/VUOERHihEPIaEp2ZB6UOV7EWh6HarnGLG+NmZHyrD+22R1AKbbdHolj506HiRH0Uw5HMz4Ko/eDfdyQJP0KO9IhjpxjyziXwx75Ci2PWc5SOAF0ytSNoqZijfWz5agRk2ka8CNoxKjN1k7zJLIgFyq9QfkGqIlrP5IHdcNh/NvVbPTzUmXe9/0HVN5BsCsCKTmPlN5kY+wmDK9435jFkT26Lpc9/vYshnK4J9AjZz3qxu+OjFgyK7MnI026JJLdCsAtiY57MthlK781cBaJ3kNgRsxTx1s/kqYPTJp/88loqmPah+QciS1bZVALziSZIX3iyiv/gHCv4hMRXzUv/GYdbwtvX5Zj25OviKwJQRqOSb8urUsS0agUq+RS9vndyUETgAmGAFQ7sbfRsAAAAASUVORK5CYII=\" width=\"111.5\" height=\"18\" style=\"width: 111.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 182.033px 8px; transform-origin: 182.033px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (note this last formula applies not only to prime numbers). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 299.367px 8px; transform-origin: 299.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website : \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eSite de Gérard Villemin\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 233.5px 8px; transform-origin: 233.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYour goal here is to check this formula for the first primes less or equal to a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.4917px 8px; transform-origin: 45.4917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003egiven integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8.16667px 8px; transform-origin: 8.16667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 97.8417px 8px; transform-origin: 97.8417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(input). To do so you will simply compute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.575px 8px; transform-origin: 211.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector. For the two first cases where the formula doesn’t apply -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e3\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.9083px 8px; transform-origin: 94.9083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e- your function should return 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 20\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 82.6333px 8px; transform-origin: 82.6333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [0, 0, 1, 2, 5, 7, 12, 15]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.4333px 8px; transform-origin: 26.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.075px 8px; transform-origin: 46.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, 7^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.75px 8px; transform-origin: 15.75px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.4px 8px; transform-origin: 7.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e11\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.4333px 8px; transform-origin: 26.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, etc…\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 40\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 133.575px 8px; transform-origin: 133.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 161.05px 8px; transform-origin: 161.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(my file exchange contributions using this equation) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/171714-isprime-optimus-fast?s_tid=srchtitle\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eIsprime optimus [FAST]\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/171709-prime-numbers-spiral?s_tid=prof_contriblnk\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime number spiral (my corresponding contribution on my file exchange)\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function k = golden_prime_squares(m)\r\n\r\n  k = m;\r\n  \r\nend","test_suite":"%%\r\nm = 4;\r\nk_correct = [0, 0];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%%\r\nm = 20;\r\nk_correct = [0, 0, 1, 2, 5, 7, 12, 15];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%%\r\nm = 40;\r\nk_correct = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('golden_prime_squares.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:55:57.000Z","deleted_by":null,"deleted_at":null,"solvers_count":23,"test_suite_updated_at":"2025-07-09T05:55:18.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-27T16:53:40.000Z","updated_at":"2026-05-17T11:22:43.000Z","published_at":"2025-06-27T17:08:17.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn december 2023, I / \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/profile/authors/8668631?s_tid=user_nav_comm\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eNicolas Douillet\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e was working on prime squares properties and I found* the formula :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{p}, p \\\\geq 5 \\\\Rightarrow \\\\exists k \\\\in \\\\mathbb{N},  \\\\mathbf{p^2 = 24k + 25}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                                   \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich I named the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e‘Christmas eve equation'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. From it I then wrote :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{p}, p \\\\geq 5 \\\\Rightarrow \\\\exists k \\\\in \\\\mathbb{N}^*,  \\\\mathbf{p^2 = 24k + 1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand I named it the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e‘golden prime squares equation’\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. These two formulas, which are actually the same one, can easily be proven starting from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = 6n \\\\pm 1, p \\\\geq 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (note this last formula applies not only to prime numbers). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website : \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSite de Gérard Villemin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour goal here is to check this formula for the first primes less or equal to a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egiven integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(input). To do so you will simply compute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector. For the two first cases where the formula doesn’t apply -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e- your function should return 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 20\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [0, 0, 1, 2, 5, 7, 12, 15]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, 7^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e11\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, etc…\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 40\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/fileexchange/171714-isprime-optimus-fast?s_tid=srchtitle\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIsprime optimus [FAST]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/fileexchange/171709-prime-numbers-spiral?s_tid=prof_contriblnk\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime number spiral (my corresponding contribution on my file exchange)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1,"title":"Times 2 - START HERE","description":"Try out this test problem first.\r\n\r\nGiven the variable x as your input, multiply it by two and put the result in y.\r\n\r\nExamples:\r\n\r\n Input  x = 2\r\n Output y is 4\r\n\r\n Input  x = 17\r\n Output y is 34\r\n","description_html":"\u003cp\u003eTry out this test problem first.\u003c/p\u003e\u003cp\u003eGiven the variable x as your input, multiply it by two and put the result in y.\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cpre\u003e Input  x = 2\r\n Output y is 4\u003c/pre\u003e\u003cpre\u003e Input  x = 17\r\n Output y is 34\u003c/pre\u003e","function_template":"function y = times2(x) % Do not edit this line.\r\n\r\n  % Modify the line below so that the output y is twice the incoming value x\r\n\r\n  y = x;\r\n\r\n  % After you modify the code, press the \"Submit\" button, and you're on your way.\r\n\r\nend % Do not edit this line.","test_suite":"%%\r\nassert(isequal(times2(1),2));\r\n\r\n%%\r\nassert(isequal(times2(11),22));\r\n\r\n%%\r\nassert(isequal(times2(-3),-6));\r\n\r\n%%\r\nassert(isequal(times2(29),58));","published":true,"deleted":false,"likes_count":2305,"comments_count":151,"created_by":1,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":116365,"test_suite_updated_at":"2012-01-25T22:41:49.000Z","rescore_all_solutions":false,"group_id":2,"created_at":"2012-01-18T01:00:16.000Z","updated_at":"2026-07-20T23:23:59.000Z","published_at":"2012-01-18T01:00:16.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTry out this test problem first.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the variable x as your input, multiply it by two and put the result in y.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  x = 2\\n Output y is 4\\n\\n Input  x = 17\\n Output y is 34]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":45960,"title":"Set the Euler-Mascheroni constant with an identity using available MATLAB functions","description":"Cody Problem \u003chttps://www.mathworks.com/matlabcentral/cody/problems/42826 42826\u003e asked us to estimate the \u003chttps://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant Euler-Mascheroni constant\u003e gamma = 0.5772156649... by summing terms in an infinite series. This problem asks you to set the constant with an identity that uses functions available in MATLAB. For example, when I started programming, I read some code that set pi not with an assignment such as pi = 3.14159 but with the identity pi = 4*atan(1).\r\n\r\nFind an analogous identity and use it to set gamma to 10 decimal places. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114.9px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57.45px; transform-origin: 407px 57.45px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84.9px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42.45px; text-align: left; transform-origin: 384px 42.45px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.35px 7.8px; transform-origin: 44.35px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCody Problem\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/42826\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e42826\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.7833px 7.8px; transform-origin: 77.7833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asked us to estimate the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eEuler-Mascheroni constant\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eγ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.783px 7.8px; transform-origin: 134.783px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e= 0.5772156649... by summing terms in an infinite series. This problem asks you to set the constant with an identity that uses functions available in MATLAB. For example, when I started programming, I read some code that set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 101.517px 7.8px; transform-origin: 101.517px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e not with an assignment such as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.2px 7.8px; transform-origin: 46.2px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 46.2px 8.25px; transform-origin: 46.2px 8.25px; \"\u003epi = 3.14159\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.8px; transform-origin: 28px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e but with the identity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.9px 7.8px; transform-origin: 53.9px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 53.9px 8.25px; transform-origin: 53.9px 8.25px; \"\u003epi = 4*atan(1)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.983px 7.8px; transform-origin: 134.983px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFind an analogous identity and use it to set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eγ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.6833px 7.8px; transform-origin: 67.6833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to 10 decimal places.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function gamma = EulerMascheroni\r\n  gamma = ...;\r\nend","test_suite":"%%\r\ny_correct = 0.5772156649;\r\nassert(isequal(EulerMascheroni,y_correct))\r\n\r\n%%\r\nfiletext = fileread('EulerMascheroni.m');\r\nsumming = ~isempty(strfind(filetext, 'sum')) || ~isempty(strfind(filetext, 'plus')) ||...\r\n          ~isempty(strfind(filetext, '+'));\r\nassert(~summing, 'Please do not sum infinite series')\r\n\r\n%%\r\nfiletext = fileread('EulerMascheroni.m');\r\nassert(isempty(strfind(filetext, '5772156649')), 'Please do not set the constant directly (or indirectly with arithmetic')\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-16T05:45:50.000Z","updated_at":"2026-07-14T07:59:58.000Z","published_at":"2020-06-16T05:53:07.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/42826\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e42826\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asked us to estimate the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eEuler-Mascheroni constant\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"gamma\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\gamma\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e= 0.5772156649... by summing terms in an infinite series. This problem asks you to set the constant with an identity that uses functions available in MATLAB. For example, when I started programming, I read some code that set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e not with an assignment such as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epi = 3.14159\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e but with the identity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epi = 4*atan(1)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind an analogous identity and use it to set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"gamma\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\gamma\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to 10 decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":61420,"title":"Sum of proper divisors (aliquot sum)","description":"A proper divisor of a positive integer n is any divisor of n except n itself. The aliquot sum is the sum of all proper divisors.\r\n\r\nGiven a positive integer n, return s, the sum of its proper divisors.\r\n\r\nFor example, the proper divisors of 12 are 1, 2, 3, 4, and 6, so the aliquot sum is 16.\r\n\r\nNote: the aliquot sum of 1 is 0 (it has no proper divisors).\r\n\r\nExamples:\r\n  n = 12  -\u003e  16   (1 + 2 + 3 + 4 + 6)\r\n  n = 6   -\u003e  6    (1 + 2 + 3, so 6 is a \"perfect number\")\r\n  n = 1   -\u003e  0\r\n  n = 7   -\u003e  1    (prime, only proper divisor is 1)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 423px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 244px 211.5px; transform-origin: 244px 211.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 21px; text-align: left; transform-origin: 220px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA proper divisor of a positive integer n is any divisor of n except n itself. The aliquot sum is the sum of all proper divisors.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a positive integer n, return s, the sum of its proper divisors.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 21px; text-align: left; transform-origin: 220px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, the proper divisors of 12 are 1, 2, 3, 4, and 6, so the aliquot sum is 16.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNote: the aliquot sum of 1 is 0 (it has no proper divisors).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eExamples:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  n = 12  -\u0026gt;  16   (1 + 2 + 3 + 4 + 6)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  n = 6   -\u0026gt;  6    (1 + 2 + 3, so 6 is a \"perfect number\")\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  n = 1   -\u0026gt;  0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  n = 7   -\u0026gt;  1    (prime, only proper divisor is 1)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = aliquot(n)\r\n    s = n;\r\nend","test_suite":"%%\r\nassert(isequal(aliquot(12), 16))\r\n%%\r\nassert(isequal(aliquot(6), 6))\r\n%%\r\nassert(isequal(aliquot(1), 0))\r\n%%\r\nassert(isequal(aliquot(7), 1))\r\n%%\r\nassert(isequal(aliquot(28), 28))   % 1+2+4+7+14, another perfect number\r\n%%\r\nassert(isequal(aliquot(100), 117))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":5142412,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2026-07-08T14:10:26.000Z","updated_at":"2026-07-18T00:22:34.000Z","published_at":"2026-07-08T14:10:26.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA proper divisor of a positive integer n is any divisor of n except n itself. The aliquot sum is the sum of all proper divisors.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a positive integer n, return s, the sum of its proper divisors.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, the proper divisors of 12 are 1, 2, 3, 4, and 6, so the aliquot sum is 16.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNote: the aliquot sum of 1 is 0 (it has no proper divisors).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e  n = 12  -\u0026gt;  16   (1 + 2 + 3 + 4 + 6)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e  n = 6   -\u0026gt;  6    (1 + 2 + 3, so 6 is a \\\"perfect number\\\")\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e  n = 1   -\u0026gt;  0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e  n = 7   -\u0026gt;  1    (prime, only proper divisor is 1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":45224,"title":"Wythoff Sequence","description":"\r\nFind the lower Wythoff sequence up to n.\r\n\r\nFor more information, \u003chttps://oeis.org/A000201\u003e","description_html":"\u003cp\u003eFind the lower Wythoff sequence up to n.\u003c/p\u003e\u003cp\u003eFor more information, \u003ca href = \"https://oeis.org/A000201\"\u003ehttps://oeis.org/A000201\u003c/a\u003e\u003c/p\u003e","function_template":"function y=wythoff(n)\r\n  y = x;\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = 1;\r\nassert(isequal(wythoff(n),y_correct))\r\n%%\r\nn = 10;\r\ny_correct = [1,3,4,6,8,9,11,12,14,16];\r\nassert(isequal(wythoff(n),y_correct))\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":363598,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2019-12-04T12:02:31.000Z","updated_at":"2026-05-30T02:36:18.000Z","published_at":"2019-12-04T12:20:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the lower Wythoff sequence up to n.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor more information,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://oeis.org/A000201\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;https://oeis.org/A000201\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":55275,"title":"List the semiprimes","description":"A semiprime number—or a 2-almost prime—is the product of two prime numbers. The numbers  and  are semiprimes, but  and  are not. The semiprimes appear in three of Ramon Villamangca’s “Easy Sequences”: Cody Problems 52859, 52990, and 53740. \r\nWrite a function to list the semiprime numbers less than or equal to the input number. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297.167px 8px; transform-origin: 297.167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA semiprime number—or a 2-almost prime—is the product of two prime numbers. The numbers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"6 = 2x3\" style=\"width: 60.5px; height: 18px;\" width=\"60.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"9 = 3x3\" style=\"width: 60.5px; height: 18px;\" width=\"60.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are semiprimes, but \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"8 = 2x2x2\" style=\"width: 84px; height: 18px;\" width=\"84\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"30 = 2x3x5\" style=\"width: 91.5px; height: 18px;\" width=\"91.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 205.517px 8px; transform-origin: 205.517px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are not. The semiprimes appear in three of Ramon Villamangca’s “Easy Sequences”: Cody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/52859\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e52859\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/52990\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e52990\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/53740\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e53740\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 264.75px 8px; transform-origin: 264.75px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to list the semiprime numbers less than or equal to the input number. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = semiprimes(n)\r\n  s = primes(n)/2;\r\nend","test_suite":"%%\r\nn = 100; \r\ns = semiprimes(n);\r\ns_correct = [4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 58 62 65 69 74 77 82 85 86 87 91 93 94 95];\r\nassert(isequal(s,s_correct))\r\n\r\n%%\r\nn = 1000; \r\ns = semiprimes(n);\r\ns_correct = [4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 58 62 65 69 74 77 82 85 86 87 91 93 94 95 106 111 115 118 119 121 122 123 129 133 134 141 142 143 145 146 155 158 159 161 166 169 177 178 183 185 187 194 201 202 203 205 206 209 213 214 215 217 218 219 221 226 235 237 247 249 253 254 259 262 265 267 274 278 287 289 291 295 298 299 301 302 303 305 309 314 319 321 323 326 327 329 334 335 339 341 346 355 358 361 362 365 371 377 381 382 386 391 393 394 395 398 403 407 411 413 415 417 422 427 437 445 446 447 451 453 454 458 466 469 471 473 478 481 482 485 489 493 497 501 502 505 511 514 515 517 519 526 527 529 533 535 537 538 542 543 545 551 553 554 559 562 565 566 573 579 581 583 586 589 591 597 611 614 622 623 626 629 633 634 635 649 655 662 667 669 671 674 679 681 685 687 689 694 695 697 698 699 703 706 707 713 717 718 721 723 731 734 737 745 746 749 753 755 758 763 766 767 771 778 779 781 785 789 791 793 794 799 802 803 807 813 815 817 818 831 835 838 841 842 843 849 851 862 865 866 869 871 878 879 886 889 893 895 898 899 901 905 913 914 917 921 922 923 926 933 934 939 943 949 951 955 958 959 961 965 973 974 979 982 985 989 993 995 998];\r\nassert(isequal(s,s_correct))\r\n\r\n%%\r\nn = 10000; \r\ns = semiprimes(n);\r\nlen_correct = 2625;\r\nsum_correct = 12736914;\r\nvar_correct = 8.447173943104530e+06;\r\nassert(isequal(length(s),len_correct) \u0026\u0026 isequal(sum(s),sum_correct) \u0026\u0026 abs(var(s)-var_correct)\u003c1e-8)\r\n\r\n%%\r\nn = 100000; \r\ns = semiprimes(n);\r\nlen_correct = 23378;\r\nsum_correct = 1138479765;\r\nvar_correct = 8.471797671132822e+08;\r\nassert(isequal(length(s),len_correct) \u0026\u0026 isequal(sum(s),sum_correct) \u0026\u0026 abs(var(s)-var_correct)\u003c1e-6)\r\n\r\n%%\r\nn = 800000; \r\ns = semiprimes(n);\r\nlen_correct = 169660;\r\nsum_correct = 66262251604;\r\nvar_correct = 5.417425253731966e+10;\r\nassert(isequal(length(s),len_correct) \u0026\u0026 isequal(sum(s),sum_correct) \u0026\u0026 abs(var(s)-var_correct)\u003c1e-4)\r\n\r\n%%\r\nfiletext = fileread('semiprimes.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'switch') || contains(filetext,'regexp'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-08-02T01:42:35.000Z","deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-08-02T01:34:03.000Z","updated_at":"2026-05-31T02:07:22.000Z","published_at":"2022-08-02T01:42:35.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA semiprime number—or a 2-almost prime—is the product of two prime numbers. The numbers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"6 = 2x3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e6 = 2\\\\times 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"9 = 3x3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e9 = 3 \\\\times 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are semiprimes, but \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"8 = 2x2x2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e8 = 2\\\\times2 \\\\times 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"30 = 2x3x5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e30 = 2\\\\times 3 \\\\times 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are not. The semiprimes appear in three of Ramon Villamangca’s “Easy Sequences”: Cody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/52859\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e52859\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/52990\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e52990\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/53740\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e53740\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to list the semiprime numbers less than or equal to the input number. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51570,"title":"Count numbers whose reciprocals' decimal expansions terminate","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 63px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 31.5px; transform-origin: 407px 31.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 378.35px 7.91667px; transform-origin: 378.35px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSome numbers (such as 4) have reciprocals with terminating decimal expansions, while others (such as 7) do not. Write a function that counts the numbers less than or equal to the input number whose reciprocals have terminating decimal expansions. For example, given 10, your function should return a count of 6, which corresponds to 1, 2, 4, 5, 8, and 10. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.1083px 7.91667px; transform-origin: 14.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e         \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = terminatingRecip(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 10;\r\ny_correct = 6;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 127;\r\ny_correct = 16;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 39841;\r\ny_correct = 61;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 63521;\r\ny_correct = 66;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 52123456;\r\ny_correct = 161;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 69871234;\r\ny_correct = 166;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 4031321321987654;\r\ny_correct = 616;\r\nassert(isequal(terminatingRecip(n),y_correct))","published":true,"deleted":false,"likes_count":5,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-27T03:16:08.000Z","updated_at":"2026-07-15T12:23:16.000Z","published_at":"2021-04-27T03:19:01.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSome numbers (such as 4) have reciprocals with terminating decimal expansions, while others (such as 7) do not. Write a function that counts the numbers less than or equal to the input number whose reciprocals have terminating decimal expansions. For example, given 10, your function should return a count of 6, which corresponds to 1, 2, 4, 5, 8, and 10. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e         \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":47803,"title":"Check if the input is a strong prime","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 109.995px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.491px 54.9977px; transform-origin: 406.497px 54.9977px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20.5882px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10.2941px; text-align: left; transform-origin: 383.496px 10.2941px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMore information about strong primes on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Strong_prime\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eWikipedia\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 79.4118px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 403.493px 39.7059px; transform-origin: 403.498px 39.7059px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisStrongPrime(11) = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisStrongPrime(2) = 0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisStrongPrime(9) = 0; \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisStrongPrime(227) = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = isStrongPrime(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 2;\r\ny_correct = 0;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nstrongPrimeList = [11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499];\r\nfor idx = 1:100\r\n    x = randi([2 500]);\r\n    y_correct = 0;\r\n    if ismember(x, strongPrimeList)\r\n        y_correct = 1;\r\n    end\r\n    assert(isequal(isStrongPrime(x),y_correct))\r\nend\r\n\r\n%%\r\nx = 4673;\r\ny_correct = 1;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 4677;\r\ny_correct = 0;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 46747;\r\ny_correct = 1;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 46748;\r\ny_correct = 0;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 467317;\r\ny_correct = 1;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 467318;\r\ny_correct = 0;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":8703,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":30,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-04T16:46:35.000Z","updated_at":"2026-05-30T19:13:47.000Z","published_at":"2020-12-04T16:46:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMore information about strong primes on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Strong_prime\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eWikipedia\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[isStrongPrime(11) = 1;\\nisStrongPrime(2) = 0;\\nisStrongPrime(9) = 0; \\nisStrongPrime(227) = 1;]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57869,"title":"Identify de Polignac numbers","description":"The numbers 125 and 329 can be written as the sum of a prime and a power of 2. For example, , and . The numbers 127 and 331, which are examples of de Polignac numbers, cannot be written in this way.\r\nWrite a function that determines whether an odd number is a de Polignac number. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 36px; transform-origin: 407px 36px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297.55px 8px; transform-origin: 297.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe numbers 125 and 329 can be written as the sum of a prime and a power of 2. For example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"125 = 109+2^4\" style=\"width: 96px; height: 19px;\" width=\"96\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"329 = 73+2^8\" style=\"width: 88.5px; height: 19px;\" width=\"88.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 320.392px 8px; transform-origin: 320.392px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The numbers 127 and 331, which are examples of de Polignac numbers, cannot be written in this way.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 255.042px 8px; transform-origin: 255.042px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that determines whether an odd number is a de Polignac number. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = isdePolignac(n)\r\n  tf = isequal(n,p+2^k);\r\nend","test_suite":"%%\r\nassert(isdePolignac(1))\r\n\r\n%%\r\nassert(~isdePolignac(17))\r\n\r\n%%\r\nassert(~isdePolignac(75))\r\n\r\n%%\r\nassert(isdePolignac(127))\r\n\r\n%%\r\nassert(isdePolignac(331))\r\n\r\n%%\r\nassert(~isdePolignac(531))\r\n\r\n%%\r\nassert(isdePolignac(905))\r\n\r\n%%\r\nassert(isdePolignac(1619))\r\n\r\n%%\r\nassert(~isdePolignac(2261))\r\n\r\n%%\r\nassert(isdePolignac(7535))\r\n\r\n%%\r\nassert(~isdePolignac(10413))\r\n\r\n%%\r\nassert(isdePolignac(21453))\r\n\r\n%%\r\nassert(isdePolignac(45233))\r\n\r\n%%\r\nassert(~isdePolignac(70999))\r\n\r\n%%\r\nassert(~isdePolignac(96415))\r\n\r\n%%\r\nassert(~isdePolignac(121399))\r\n\r\n%%\r\nassert(isdePolignac(148243))\r\n\r\n%%\r\nassert(isdePolignac(172841))\r\n\r\n%%\r\nassert(isdePolignac(201599))\r\n\r\n%%\r\nassert(isdePolignac(227107))\r\n\r\n%%\r\nassert(isdePolignac(253151))\r\n\r\n%%\r\nassert(~isdePolignac(267267))\r\n\r\n%%\r\nassert(~isdePolignac(271271))\r\n\r\n%%\r\nassert(isdePolignac(273421))\r\n\r\n%%\r\nassert(isdePolignac(542459))\r\n\r\n%%\r\nassert(isdePolignac(2000039))\r\n\r\n%%\r\nassert(~isdePolignac(123456789))\r\n\r\n%%\r\nassert(isdePolignac(123456791))\r\n\r\n%%\r\nassert(isequal(bin2dec(num2str(arrayfun(@isdePolignac,5893:2:5933)')'),288))\r\n\r\n%%\r\nassert(isequal(bin2dec(num2str(arrayfun(@isdePolignac,21671:10:21791)')'),4624))\r\n\r\n%%\r\nfiletext = fileread('isdePolignac.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2023-03-29T05:27:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":24,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-03-29T05:27:08.000Z","updated_at":"2026-05-25T01:35:42.000Z","published_at":"2023-03-29T05:27:20.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe numbers 125 and 329 can be written as the sum of a prime and a power of 2. For example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"125 = 109+2^4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e125 = 109+2^4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"329 = 73+2^8\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e329 = 73+2^8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The numbers 127 and 331, which are examples of de Polignac numbers, cannot be written in this way.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that determines whether an odd number is a de Polignac number. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60782,"title":"Convert integers from primary notation to base 10 ","description":"The primary system of expressing numbers has three digits: m, x, and p, which correspond to -1, 0, and 1. The places in a primary number correspond to the prime numbers. For example, the number 6 could be ppm in primary notation because . Similarly, the number 20 could be ppxxp in primary notation. \r\nBecause integers can be expressed as sums (and differences) of primes in several ways, several primary notations for a number can exist. Cody Problem 60783 deals with proper primary notation.  \r\nWrite a function to convert integers expressed in primary notation to base 10. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 145.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 72.75px; transform-origin: 407px 72.75px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 64.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32.25px; text-align: left; transform-origin: 384px 32.25px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.5px 8px; transform-origin: 14.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.5px 8px; transform-origin: 25.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eprimary \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 150px 8px; transform-origin: 150px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esystem of expressing numbers has three digits: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003ex\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160px 8px; transform-origin: 160px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which correspond to -1, 0, and 1. The places in a primary number correspond to the prime numbers. For example, the number 6 could be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12px 8px; transform-origin: 12px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 12px 8.5px; transform-origin: 12px 8.5px; \"\u003eppm\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91.5px 8px; transform-origin: 91.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in primary notation because \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"163\" height=\"18.5\" alt=\"6 = 5*1 + 3*1 + 2*(-1)\" style=\"width: 163px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.5px 8px; transform-origin: 110.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Similarly, the number 20 could be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20px 8px; transform-origin: 20px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 20px 8.5px; transform-origin: 20px 8.5px; \"\u003eppxxp\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.5px 8px; transform-origin: 64.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in primary notation. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380px 8px; transform-origin: 380px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBecause integers can be expressed as sums (and differences) of primes in several ways, several primary notations for a number can exist. \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/60783\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 60783\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35px 8px; transform-origin: 35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eproper\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59px 8px; transform-origin: 59px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e primary notation.  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 245px 8px; transform-origin: 245px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to convert integers expressed in primary notation to base 10. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = primary2dec(s)\r\n  y = base2dec(s,'primes')\r\nend","test_suite":"%%\r\nassert(isequal(primary2dec('ppm'),6))\r\n\r\n%%\r\nassert(isequal(primary2dec('ppxxp'),20))\r\n\r\n%%\r\nassert(isequal(primary2dec('x'),0))\r\n\r\n%%\r\nassert(isequal(primary2dec('pm'),1))\r\n\r\n%%\r\nassert(isequal(primary2dec('mp'),-1))\r\n\r\n%%\r\nassert(isequal(primary2dec('pxxpp'),16))\r\n\r\n%%\r\nassert(isequal(primary2dec('pxppxp'),27))\r\n\r\n%%\r\nassert(isequal(primary2dec('ppppp'),28))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmmmpm'),-35))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmxmmxmm'),-59))\r\n\r\n%%\r\nassert(isequal(primary2dec('pppppppm'),73))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmmmxmmmmm'),-116))\r\n\r\n%%\r\nassert(isequal(primary2dec('ppppppppppppm'),234))\r\n\r\n%%\r\nassert(isequal(primary2dec('pppppxppppppppppppxp'),589))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmmmmmmmxmmmmmmmmmmmmmmmmmmmxmmx'),-1753))\r\n\r\n%%\r\nassert(isequal(primary2dec('pppppppxpppppppppppppppppppppppppppppppppppppppxppppp'),5618))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmxmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmxmxm'),-74921))\r\n\r\n%%\r\nassert(isequal(primary2dec('ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppxpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppxpp'),514888))\r\n\r\n%%\r\nfiletext = fileread('primary2dec.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'str2num'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2025-01-06T21:00:45.000Z","deleted_by":null,"deleted_at":null,"solvers_count":11,"test_suite_updated_at":"2025-01-02T00:40:47.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-12-31T17:58:19.000Z","updated_at":"2026-06-06T01:59:11.000Z","published_at":"2024-12-31T17:58:24.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprimary \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esystem of expressing numbers has three digits: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, which correspond to -1, 0, and 1. The places in a primary number correspond to the prime numbers. For example, the number 6 could be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eppm\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in primary notation because \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"6 = 5*1 + 3*1 + 2*(-1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e6 = 5\\\\cdot 1 + 3\\\\cdot 1 + 2\\\\cdot(-1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Similarly, the number 20 could be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eppxxp\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in primary notation. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBecause integers can be expressed as sums (and differences) of primes in several ways, several primary notations for a number can exist. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/60783\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 60783\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eproper\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e primary notation.  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to convert integers expressed in primary notation to base 10. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52060,"title":"Identify Ruth-Aaron numbers","description":"After Hank Aaron hit his 715th home run and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers ( and ) comprises all of the prime numbers up to 17. \r\nHe told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*\r\nWrite a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.\r\n*The story continues in a Numberphile video on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 237px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 118.5px; transform-origin: 407px 118.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.7167px 7.91667px; transform-origin: 16.7167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAfter \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.youtube.com/watch?v=QjqYThEVoSQ\"\u003e\u003cspan style=\"perspective-origin: 70.8px 7.91667px; transform-origin: 70.8px 7.91667px; \"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eHank Aaron hit his 715\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"perspective-origin: 5.83333px 7.91667px; transform-origin: 5.83333px 7.91667px; \"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eth\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"perspective-origin: 31.5083px 7.91667px; transform-origin: 31.5083px 7.91667px; \"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003e home run\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 259.142px 7.91667px; transform-origin: 259.142px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"714 = 2*3*7*17\" style=\"width: 114.5px; height: 18px;\" width=\"114.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAM8AAAAkCAYAAAAuAm4eAAADrUlEQVR4nO2b3bXqIBCFdw/pwAbSQCqwAjuwAzuwBWuwhNODLaSGtHDvA5klRkkYGAR1f2vxZCDzw8AwQYAQQgghhBBCCCHkS+jm9k3sagtAbDgAOEW2g2JMzQRZk+GGzwueAWF9rgD29UQDoPePVd9SaGXqARxxn9NJ+nQAJgD/IttpY7wDgHF+to+UYbfxzku0Nu0gNnjVxopypfjHom8ptDL1CPtma26/fLk49G+lyQtCETq8ECrWwBe4AA69u7VVbguxaUifGrtOjn9y+pYiRaYe93m2n8c4LsY5aoQYAZw3npGd4Rb4fT+3Di4l0RhYxq6dxlgSY9N3kuOfnL6lSJVpxOtjR4d7AIXm+BMDXBRucUT8tnaCzsBnOME/7UwTYo/6k2sNrX+s+pYiViY5U4c4z2NMmpfHpEQ3xBtMY2A5b41zP1lNPhlJcbMOogX51eDpsT63ZJyYzSQaSatiD7kaA/vPLgsEQ4qwlRnwWp8/xFcpS/OrwbOFpH+mxwdJ2WJzeI0yfr5qUv2ozAnrlcsr6u+sDJ5neujmeDSSssXuBCnKdHARf4FNAO0Q/91qreXsfgOcM5bBZJoWJMDgeWSA85H5wqZN2YB8ZXZ4LI2njBNKn7RNVbYM0OF+GLUcNxUGj2PA82I9wTBt06ZsgJ2BfcW026nVzmM5QeTbj6ocWgAGz+Ptj+VCrcmyVtGmbICdgZNq740jZ7wJ+qskI2xWxhaCpxV9hGW2kz3fUlI2wHZ1Sqq9N4x8A9Laxi+s5F7taSF4WtHHx1+stYvbEykpG2AbPHJ2+ZbgAdJs49v0mvn+FoKnFX2WyJzPHkuiUJv/lQge7TbaQrUthNhGW9k5zC23ItRC8ABt6LPELzQl7zxS907ZUq0NnLL7tVRt89m6I/gOWgkeK0oET1YqKWeNlI9GlsrIIU47TovVNuBuG5aq7SiRtmV9nJeULaUSEquMlAtD11ZkFbgkyFADCdgjXqcico+v9h/7fjV4Org5FUrHpGCQ5R9J2abEQfwPgmvK+OnR3+JZ+ep7SZShBn6pc8TjwrODc8oN9S+KxvrHum8pYmXyK3wnPM6rDs43IzL9I8JoV/wObhfxr6NcV4TxVww/iESJVi5SxrLH81WcG5xOE54d9m60/rHqW4rc+eb/+XJEOGNQIf970KwsW2eMUI6/w/1/5JLCfeJtakEc6us9oP7umeqf3L6lSJWpXzy3Rzs7KCGEEEIIIYQQQgghhBBCCCGEtMV/m09p4GPXQvwAAAAASUVORK5CYII=\" alt=\"715 = 5*11*13\" style=\"width: 103.5px; height: 18px;\" width=\"103.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 144.3px 7.91667px; transform-origin: 144.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) comprises all of the prime numbers up to 17. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.158px 7.91667px; transform-origin: 383.158px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHe told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 365.35px 7.91667px; transform-origin: 365.35px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 78.9583px 7.91667px; transform-origin: 78.9583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e*The story continues in a \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.youtube.com/watch?v=aCq04N9it8U\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eNumberphile video\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 232.967px 7.91667px; transform-origin: 232.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = RuthAaron(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 5;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 25;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 51;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 78;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 101;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 153;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 222;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 493;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 713;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 714;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 715;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 716;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 24024;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 154843;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 977779;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 6905998;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 29817473;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 89460293;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 89460294;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 89460295;\r\nassert(isequal(RuthAaron(n),'RA'))\r\n\r\n%%\r\nn = 89460296;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%  \r\nn = 89460297;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 144426976;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 333333333;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 715159952;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 888681984;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 1925922590;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 1925922591;\r\nassert(isequal(RuthAaron(n),'A'))","published":true,"deleted":false,"likes_count":3,"comments_count":3,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":29,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-11T02:03:10.000Z","updated_at":"2026-05-25T05:46:26.000Z","published_at":"2021-06-11T02:14:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAfter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/watch?v=QjqYThEVoSQ\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHank Aaron hit his 715\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eth\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e home run\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"714 = 2*3*7*17\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e714 = 2\\\\cdot 3\\\\cdot 7 \\\\cdot 17\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"715 = 5*11*13\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e715 = 5\\\\cdot 11 \\\\cdot 13\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) comprises all of the prime numbers up to 17. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHe told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e*The story continues in a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/watch?v=aCq04N9it8U\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNumberphile video\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":61322,"title":"Determine if a number is prime","description":"Given a positive integer n, return 1 if it is prime, 0 otherwise. Example: is_prime(7) returns 1, is_prime(4) returns 0","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 42.4px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 244px 21.2px; transform-origin: 244px 21.2px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 21.2px; text-align: left; transform-origin: 220px 21.2px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a positive integer n, return 1 if it is prime, 0 otherwise. Example: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eis_prime(7)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eis_prime(4)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function result = is_prime(n)\r\n  result = 0;\r\nend","test_suite":"%% Test 1\r\nassert(isequal(is_prime(7), 1))\r\n%% Test 2\r\nassert(isequal(is_prime(4), 0))\r\n%% Test 3\r\nassert(isequal(is_prime(2), 1))\r\n%% Test 4\r\nassert(isequal(is_prime(1), 0))\r\n%% Test 5\r\nassert(isequal(is_prime(13), 1))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":5047536,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":24,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2026-05-04T22:55:10.000Z","updated_at":"2026-07-20T09:19:27.000Z","published_at":"2026-05-04T22:55:10.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a positive integer n, return 1 if it is prime, 0 otherwise. Example: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eis_prime(7)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eis_prime(4)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44229,"title":"How brilliant are you?","description":"A Brilliant number is defined as a number with two prime factors, both of which have the same number of digits.  Some examples:\r\n\r\n10=2*5.  Since 2 and 5 have the same number of digits, 10 is a brilliant number.\r\n\r\n22=2*11.  Although there are two prime factors, they have a different number of digits, so 22 is not a brilliant number.\r\n\r\n30=2*3*5.  Although each prime factor has the same number of digits, there are more than two of them, so 30 is not a brilliant number.\r\n\r\nGiven a number, write a MATLAB script to determine if the number is brilliant or not.","description_html":"\u003cp\u003eA Brilliant number is defined as a number with two prime factors, both of which have the same number of digits.  Some examples:\u003c/p\u003e\u003cp\u003e10=2*5.  Since 2 and 5 have the same number of digits, 10 is a brilliant number.\u003c/p\u003e\u003cp\u003e22=2*11.  Although there are two prime factors, they have a different number of digits, so 22 is not a brilliant number.\u003c/p\u003e\u003cp\u003e30=2*3*5.  Although each prime factor has the same number of digits, there are more than two of them, so 30 is not a brilliant number.\u003c/p\u003e\u003cp\u003eGiven a number, write a MATLAB script to determine if the number is brilliant or not.\u003c/p\u003e","function_template":"function y = brilliant(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nassert(isequal(brilliant(4),1))\r\n%%\r\nassert(isequal(brilliant(8),0))\r\n%%\r\nassert(isequal(brilliant(40),0))\r\n%%\r\nassert(isequal(brilliant(343),0))\r\n%%\r\nassert(isequal(brilliant(1536),0))\r\n%%\r\nassert(isequal(brilliant(1537),1))\r\n%%\r\nassert(isequal(brilliant(49165),0))\r\n%%\r\nassert(isequal(brilliant(657721),1))\r\n%%\r\nassert(isequal(brilliant(768819),0))\r\n%%\r\nassert(isequal(brilliant(13717421),1))\r\n%%\r\nassert(isequal(brilliant(123456789),0))\r\n%%\r\nassert(isequal(brilliant(669562601),1))\r\n%%\r\nassert(isequal(brilliant(1234567890),0))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":66,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2017-06-01T18:34:04.000Z","updated_at":"2026-02-24T14:03:59.000Z","published_at":"2017-06-01T18:34:04.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA Brilliant number is defined as a number with two prime factors, both of which have the same number of digits. Some examples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e10=2*5. Since 2 and 5 have the same number of digits, 10 is a brilliant number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e22=2*11. Although there are two prime factors, they have a different number of digits, so 22 is not a brilliant number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e30=2*3*5. Although each prime factor has the same number of digits, there are more than two of them, so 30 is not a brilliant number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a number, write a MATLAB script to determine if the number is brilliant or not.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":47798,"title":"Check if the input is a balanced prime","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 90.1425px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.491px 45.0712px; transform-origin: 406.497px 45.0712px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20.5882px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10.2941px; text-align: left; transform-origin: 383.496px 10.2941px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMore about Balanced Primes on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Balanced_prime\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eWikipedia\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 59.5588px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 403.493px 29.7794px; transform-origin: 403.498px 29.7794px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisBalancedPrime(3) = false;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisBalancedPrime(7) = false;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisBalancedPrime(5) = true; \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration: none; text-decoration-color: rgb(2, 128, 9); \"\u003e% since  (3+7)/2 = 5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = isBalancedPrime(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 2;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n%%\r\nx = 3;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n%%\r\nx = 5;\r\ny_correct = 1;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n%%\r\nx = 9;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nlist = [5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103];\r\nfor idx = 1:50\r\n    x = randi([10 1103]);\r\n        if ismember(x,list)\r\n            y_correct = 1;\r\n        else\r\n            y_correct = 0;\r\n        end\r\n    assert(isequal(isBalancedPrime(x),y_correct))\r\nend\r\n\r\n\r\n%%\r\nx = 81001;\r\ny_correct = 1;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 81402;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 177487;\r\ny_correct = 1;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 467101;\r\ny_correct = 1;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 467102;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":8703,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":30,"test_suite_updated_at":"2020-12-04T16:29:03.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-04T16:26:34.000Z","updated_at":"2026-05-30T19:13:44.000Z","published_at":"2020-12-04T16:26:34.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMore about Balanced Primes on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Balanced_prime\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eWikipedia\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[isBalancedPrime(3) = false;\\nisBalancedPrime(7) = false;\\nisBalancedPrime(5) = true; % since  (3+7)/2 = 5]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44748,"title":"Amicable numbers","description":"Test whether two numbers are \u003chttps://en.wikipedia.org/wiki/Amicable_numbers amicable\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\r\n\r\n\r\nExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.","description_html":"\u003cp\u003eTest whether two numbers are \u003ca href = \"https://en.wikipedia.org/wiki/Amicable_numbers\"\u003eamicable\u003c/a\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\u003c/p\u003e\u003cp\u003eExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.\u003c/p\u003e","function_template":"function y = amicable(m,n)\r\n  y = false;\r\nend","test_suite":"%%\r\nm = 220; n = 284;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 220; n = 504;\r\ny_correct = false;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 2620; n = 2924;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 5020; n = 5564;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 2924; n = 5020;\r\ny_correct = false;\r\nassert(isequal(amicable(m,n),y_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":4,"created_by":254267,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":80,"test_suite_updated_at":"2018-10-22T18:10:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-10-22T17:57:52.000Z","updated_at":"2026-06-08T05:35:58.000Z","published_at":"2018-10-22T18:02:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTest whether two numbers are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Amicable_numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eamicable\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60972,"title":"Check Bertrand-Chebyshev's theorem about prime numbers","description":"Historical context\r\nBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than 1, there always exists at least on prime number between it and its double : \r\n\r\n\r\n\r\nProblem statement\r\nFor a given input n, List the prime number(s) p in the interval ]n; 2n[.\r\n\r\nExamples\r\n\r\nn = 2 =\u003e p = 3;\r\nn = 3 =\u003e p = 5;\r\nn = 7 =\u003e  p = [11, 13];\r\nn = 47 =\u003e p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 697.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 348.733px; transform-origin: 408px 348.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 385px 8px; transform-origin: 385px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 223.658px 8px; transform-origin: 223.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethere always exists at least on prime number between it and its double : \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; 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4ZzqN0yWHGoepV+Us28EyJid++zV7veNaqJdEN+l6q5dK6srhjjXxaf1QoK5xrH6vdEEQr3dZOQRgF2DGsoFAoHAeASC+MZjGiUGAoFAIBAI7BiBIL4dd06oFggEAoFAIDAegSC+8ZhGiYFAIBAIBAI7RuA/zuJmZAlv4pMAAAAASUVORK5CYII=\" width=\"223\" height=\"19.5\" style=\"width: 223px; height: 19.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.0667px 8px; transform-origin: 54.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given input \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.675px 8px; transform-origin: 81.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, List the prime number(s) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.8917px 8px; transform-origin: 66.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the interval ]n; 2n[.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.6833px 8px; transform-origin: 46.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 2 =\u0026gt; p = 3;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.6833px 8px; transform-origin: 46.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 3 =\u0026gt; p = 5;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 68.3333px 8px; transform-origin: 68.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 7 =\u0026gt;  p = [11, 13];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 152.467px 8px; transform-origin: 152.467px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 47 =\u0026gt; p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = check_Bertrand_Chebyshev_theorem(n)\r\n    p = n;\r\nend","test_suite":"%%\r\nn = 2;\r\np_correct = 3;\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 3;\r\np_correct = 5;\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 7;\r\np_correct = [11, 13];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 11;\r\np_correct = [13, 17, 19];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 47;\r\np_correct = [53, 59, 61, 67, 71, 73, 79, 83, 89];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_Bertrand_Chebyshev_theorem.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:03:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":41,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:28:51.000Z","updated_at":"2026-07-20T06:40:55.000Z","published_at":"2025-07-21T04:30:08.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethere always exists at least on prime number between it and its double : \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall n \\\\in \\\\mathbb{N}^* - \\\\{ 1 \\\\}, \\\\exists p \\\\in \\\\mathbb{P}, \\\\mathbf{n \u0026lt; p \u0026lt; 2n}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given input \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, List the prime number(s) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in the interval ]n; 2n[.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 2 =\u0026gt; p = 3;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 3 =\u0026gt; p = 5;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 7 =\u0026gt;  p = [11, 13];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 47 =\u0026gt; p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60946,"title":"Fermat primes","description":"Historical context\r\n\r\n\r\nThe french mathematician Pierre de Fermat found the formula  to find some prime numbers. He thought / conjectured all these numbers where primes, but P(5) is actually the first not prime Fermat number.\r\n\r\n\r\nProblem statement\r\n\r\nFor a given integer m, , list the Fermat prime numbers less or equal to m in a vector fp. Also compute their frequency / ratio f compare to all the primes in the same range. If  , your function should return fp = [] and f = 0.\r\n\r\nExamples\r\n\r\nIf m = 10 then fp = [3, 5] and f = 2/3 since 3 = P(0), 5 = P(1), and there are in total three primes in the range [3; 10] ;\r\nIf m = 300 then fp = [3, 5, 17, 257] and f = 4/61 since 3 = P(0), 5 = P(1), 17 = P(2), 257 = P(3) and there are in total 61 prime numbers in the range [3; 300] ; \r\nIf m = 2 then fp = [] and f = 0;\r\n\r\n\r\nTip \r\n\r\n here.\r\n\r\n\r\nFobidden functions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\n\r\n\r\nSee also\r\nProblem 60942. Euler primes\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1001.03px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 500.517px; transform-origin: 408px 500.517px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 22.5px; text-align: left; transform-origin: 385px 22.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 193.7px 8px; transform-origin: 193.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe french mathematician Pierre de Fermat found the formula \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"96\" height=\"23.5\" style=\"width: 96px; height: 23.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.625px 8px; transform-origin: 132.625px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to find some prime numbers. He thought / conjectured all these numbers where primes, but P(5) is actually the first not prime Fermat number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 144.3px 8px; transform-origin: 144.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Fermat prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.1083px 8px; transform-origin: 38.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 98.6167px 8px; transform-origin: 98.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 146.242px 8px; transform-origin: 146.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.8583px 8px; transform-origin: 82.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23.1333px 8px; transform-origin: 23.1333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e fp = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 10\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.5833px 8px; transform-origin: 28.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 2/3 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 56.0083px 8px; transform-origin: 56.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e3 = P(0), 5 = P(1),\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 169.192px 8px; transform-origin: 169.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and there are in total three primes in the range [3; 10] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2.33333px 8px; transform-origin: 2.33333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 300\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 55.8083px 8px; transform-origin: 55.8083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 17, 257]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 4/61 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 86.9333px 8px; transform-origin: 86.9333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e3 = P(0), 5 = P(1), 17 = P(2)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.875px 8px; transform-origin: 32.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e257 = P(3)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 66.125px 8px; transform-origin: 66.125px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand there are in total 61\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 31.1px 8px; transform-origin: 31.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [3; 300] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.0333px 8px; transform-origin: 20.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.25px 8px; transform-origin: 19.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003efp = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60942-euler-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60942. Euler primes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [fp, f] = fermat_primes(n)\r\n  fp = n;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 10;\r\nfp_correct = [3, 5];\r\nf_correct = 2/3;\r\n[fp,f] = fermat_primes(m);\r\nassert(isequal([fp,f],[fp_correct,f_correct]));\r\n\r\n%%\r\nm = 300;\r\nfp_correct = [3, 5, 17, 257];\r\nf_correct = 4/61;\r\n[fp,f] = fermat_primes(m);\r\nassert(isequal([fp,f],[fp_correct,f_correct]));\r\n\r\n%%\r\nm = 2;\r\nfp_correct = [];\r\nf_correct = 0;\r\n[fp,f] = fermat_primes(m);\r\nassert(isequal([fp,f],[fp_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('fermat_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:52:07.000Z","deleted_by":null,"deleted_at":null,"solvers_count":21,"test_suite_updated_at":"2025-07-09T05:57:08.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-27T05:33:53.000Z","updated_at":"2026-06-09T14:35:13.000Z","published_at":"2025-06-27T06:14:34.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe french mathematician Pierre de Fermat found the formula \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{P(n) = 2^{2^n} + 1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to find some prime numbers. He thought / conjectured all these numbers where primes, but P(5) is actually the first not prime Fermat number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em \\\\geq 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Fermat prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em \u0026lt; 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e fp = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 2/3 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3 = P(0), 5 = P(1),\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and there are in total three primes in the range [3; 10] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 300\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 17, 257]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 4/61 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3 = P(0), 5 = P(1), 17 = P(2)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e257 = P(3)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand there are in total 61\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [3; 300] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efp = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en \\\\leq 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden 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w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60942-euler-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60942. Euler primes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":47808,"title":"Check if the input is an interprime","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 70.8px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 331.5px 35.4px; transform-origin: 331.5px 35.4px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 308.5px 10.4px; text-align: left; transform-origin: 308.5px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMore information about interprimes on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Interprime\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eWikipedia\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 40px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 328.5px 20px; transform-origin: 328.5px 20px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.5px 10px; transform-origin: 328.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisInterprime(2) = 0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.5px 10px; transform-origin: 328.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisInterprime(4) = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = isInterprime(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 2;\r\ny_correct = 0;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n%%\r\ninterprimeList = [4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99];\r\nfor idx = 1:100\r\n    x = randi([2 99]);\r\n    y_correct = 0;\r\n    if ismember(x,interprimeList)\r\n        y_correct = 1;\r\n    end\r\n    assert(isequal(isInterprime(x),y_correct))\r\nend\r\n\r\n%%\r\nx = 4676;\r\ny_correct = 1;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 46725;\r\ny_correct = 1;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 4622;\r\ny_correct = 0;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n%%\r\nx = 467475;\r\ny_correct = 1;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 467476;\r\ny_correct = 0;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n%% \r\nx = 499994;\r\ny_correct = 1;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n    ","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":8703,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":"2020-12-28T11:41:15.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2020-12-04T17:23:08.000Z","updated_at":"2026-05-30T19:13:49.000Z","published_at":"2020-12-04T17:23:08.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMore information about interprimes on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Interprime\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eWikipedia\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[isInterprime(2) = 0;\\nisInterprime(4) = 1;]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":56743,"title":"Count the unitary divisors of a number","description":"Cody Problem 56738 asks for a list of the unitary divisors of a number. For this problem, write a function to count the unitary divisors of a number. For example, the unitary divisors of 18 are 1, 2, 9, and 18; therefore, the number of unitary divisors is 4.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 63px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 31.5px; transform-origin: 407px 31.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/56738-list-unitary-divisors-of-a-number\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 56738\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 318.25px 8px; transform-origin: 318.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks for a list of the unitary divisors of a number. For this problem, write a function to count the unitary divisors of a number. For example, the unitary divisors of 18 are 1, 2, 9, and 18; therefore, the number of unitary divisors is 4.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = countUDivisors(n)\r\n  y = length(n(y/n==true));\r\nend","test_suite":"%%\r\nn = 18;\r\ny_correct = 4;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 128;\r\ny_correct = 2;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 996;\r\ny_correct = 8;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 1228;\r\ny_correct = 4;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 54321;\r\ny_correct = 8;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 648207;\r\ny_correct = 8;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 840519372;\r\ny_correct = 128;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 7420738134810;\r\ny_correct = 4096;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = flintmax/2-7;\r\ny_correct = 8;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = flintmax-4;\r\ny_correct = 64;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%% \r\np = primes(randi(1000000));\r\nn = p(end);\r\nassert(isequal(countUDivisors(n),2))\r\n\r\n%%\r\nfiletext = fileread('countUDivisors.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-11-24T17:08:57.000Z","deleted_by":null,"deleted_at":null,"solvers_count":15,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-11-24T17:07:22.000Z","updated_at":"2026-07-09T14:40:21.000Z","published_at":"2022-11-24T17:08:57.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/56738-list-unitary-divisors-of-a-number\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 56738\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks for a list of the unitary divisors of a number. For this problem, write a function to count the unitary divisors of a number. For example, the unitary divisors of 18 are 1, 2, 9, and 18; therefore, the number of unitary divisors is 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":2733,"title":"Evil Number","description":"Check if a given natural number is evil or not. \r\n\r\nRead more at \u003chttps://oeis.org/A001969 OEIS\u003e.","description_html":"\u003cp\u003eCheck if a given natural number is evil or not.\u003c/p\u003e\u003cp\u003eRead more at \u003ca href = \"https://oeis.org/A001969\"\u003eOEIS\u003c/a\u003e.\u003c/p\u003e","function_template":"function tf = isevil(n)\r\n  tf = ;\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = false;\r\nassert(isequal(isevil(x),y_correct))\r\n%%\r\nx = 3;\r\ny_correct = true;\r\nassert(isequal(isevil(x),y_correct))\r\n%%\r\nx = [18, 20, 23, 24, 27, 45, 46, 48, 96, 99, 123,];\r\ny_correct = true;\r\nassert(isequal(all(arrayfun(@isevil,x)),y_correct))\r\n%%\r\nx = [14, 16, 19, 37, 38, 55, 56, 59, 62,  79, 82, 91, 93, 94, 97, 98, 117, 118, 121];\r\ny_correct = false;\r\nassert(isequal(any(arrayfun(@isevil,x)),y_correct))\r\n%%\r\nx = 2^randi([5 10])+1;\r\ny_correct = true;\r\nassert(isequal(isevil(x),y_correct))\r\n\r\n%%\r\n% more test cases may be introduced\r\n%%\r\n% DISABLED\r\n% ________'FAIR'_SCORING_SYSTEM______________\r\n%\r\n% This section scores for usage of ans\r\n% and strings, which are common methods \r\n% to reduce cody size of solution.\r\n% Here, strings are threated like vectors.\r\n% Please do not hack it, as this problem\r\n% is not mentioned to be a hacking problem.\r\n% \r\n  try\r\n% disable:\r\nassert(false) \r\n%\r\n  size_old = feval(@evalin,'caller','score');\r\n%\r\n  all_nodes = mtree('isevil.m','-file');\r\n  str_nodes = mtfind(all_nodes,'Kind','STRING');\r\n   eq_nodes = mtfind(all_nodes,'Kind','EQUALS');\r\nprint_nodes = mtfind(all_nodes,'Kind','PRINT');\r\n expr_nodes = mtfind(all_nodes,'Kind','EXPR');\r\n%\r\n       size = count(all_nodes)           ...\r\n              +sum(str_nodes.nodesize-1) ...\r\n              +2*(count(expr_nodes)      ...\r\n                  +count(print_nodes)    ...\r\n                  -count(eq_nodes));\r\n%\r\n  feval(@assignin,'caller','score',size);\r\n%\r\n  fprintf('Size in standard cody scoring is %i.\\n',size_old);\r\n  fprintf('Here it is %i.\\n',size);\r\n  end\r\n%\r\n%_________RESULT_____________________________","published":true,"deleted":false,"likes_count":3,"comments_count":6,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":276,"test_suite_updated_at":"2016-12-26T10:21:47.000Z","rescore_all_solutions":true,"group_id":8,"created_at":"2014-12-07T21:50:01.000Z","updated_at":"2026-06-08T04:55:31.000Z","published_at":"2015-01-19T12:47:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck if a given natural number is evil or not.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRead more at\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://oeis.org/A001969\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOEIS\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60942,"title":"Euler primes","description":"Historical context\r\n\r\n\r\nThe great swiss mathematician Leonhard Euler found the formula  to give some prime numbers.\r\n\r\n\r\nProblem statement\r\n\r\nFor a given integer m, m \u003e= 41, list the Euler prime numbers less or equal to m in a vector ep. Also compute their frequency / ratio f compare to all the primes in the same range. If m is not a positive integer greater or equal to 41, your function should return ep = [] and f = 0.\r\n\r\nExamples (check the tests for more)\r\n\r\nIf m = 41 then ep = 41 and f = 1 since 41 = P(0) and it is the only prime number m greater or equal and less or equal to 41;\r\nIf m = 100 then ep = [41, 43, 47, 53, 61, 71, 83, 97] and f = 8/13 since 41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), and in total there are 13 prime numbers in the range [41; 100] ; \r\nIf m = 19 then ep = [] and f = 0;\r\n\r\n\r\nTip \r\n\r\nn is less or equal to 107 here.\r\n\r\n\r\nFobidden functions\r\n\r\nregexp\r\nstr2num\r\nassignin\r\n\r\nSee also\r\nProblem 60944. Ulam primes first formula, Problem 60943. Ulam primes second formula\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 988.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 494.233px; transform-origin: 408px 494.233px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 204.608px 8px; transform-origin: 204.608px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe great swiss mathematician Leonhard Euler found the formula \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"125\" height=\"19.5\" style=\"width: 125px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.1833px 8px; transform-origin: 92.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to give some prime numbers.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.075px 8px; transform-origin: 26.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u0026gt;= 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 138.475px 8px; transform-origin: 138.475px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Euler prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.1667px 8px; transform-origin: 36.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 8px; transform-origin: 7.78333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.3833px 8px; transform-origin: 96.3833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 286.658px 8px; transform-origin: 286.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.6667px 8px; transform-origin: 11.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e41, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.35px 8px; transform-origin: 63.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.6917px 8px; transform-origin: 24.6917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e ep = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3333px 8px; transform-origin: 79.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests for more)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 51.0833px; transform-origin: 392px 51.0833px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.1417px 8px; transform-origin: 16.1417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.9833px 8px; transform-origin: 28.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e41 = P(0)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 99.575px 8px; transform-origin: 99.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and it is the only prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.533px 8px; transform-origin: 108.533px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e greater or equal and less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10.1167px 8px; transform-origin: 10.1167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e41;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 100\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 110.258px 8px; transform-origin: 110.258px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = [41, 43, 47, 53, 61, 71, 83, 97]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 8/13 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 141.192px 8px; transform-origin: 141.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 64.1833px 8px; transform-origin: 64.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand in total there are\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.6667px 8px; transform-origin: 11.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e 13 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.9917px 8px; transform-origin: 34.9917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [41; 100] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 19 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.8083px 8px; transform-origin: 20.8083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.3417px 8px; transform-origin: 58.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e107\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60944-ulam-primes-first-formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eProblem 60944. Ulam primes first formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60943-ulam-primes-second-formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eProblem 60943. Ulam primes second formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ep,f] = euler_primes(m)\r\n  ep = m;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 41;\r\nep_correct = 41;\r\nf_correct = 1;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 100;\r\nep_correct = [41, 43, 47, 53, 61, 71, 83, 97];\r\nf_correct = 8/13;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 200;\r\nep_correct = [41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197];\r\nf_correct = 13/34;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 19;\r\nep_correct = [];\r\nf_correct = 0;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('euler_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:49:26.000Z","deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":"2025-07-09T05:56:34.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-26T14:44:30.000Z","updated_at":"2026-05-16T04:50:50.000Z","published_at":"2025-06-26T15:44:04.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe great swiss mathematician Leonhard Euler found the formula \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{P(n) = n^2 + n + 41}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to give some prime numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u0026gt;= 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Euler prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e ep = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41 = P(0)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and it is the only prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e greater or equal and less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 100\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = [41, 43, 47, 53, 61, 71, 83, 97]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 8/13 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand in total there are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e 13 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [41; 100] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 19 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e107\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e here.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60944-ulam-primes-first-formula\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem 60944. Ulam primes first formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60943-ulam-primes-second-formula\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem 60943. Ulam primes second formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60997,"title":"Multiply real numbers to get the smallest integers","description":"Write a function that takes a vector of real numbers and multiplies them by a factor to produce the integers with the smallest absolute value. For example, if the input vector is [3/4 5/6 7/10], the output should be [45 50 42]; that is, the input vector should be multiplied by 60.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 63px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 31.5px; transform-origin: 408px 31.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes a vector of real numbers and multiplies them by a factor to produce the integers with the smallest absolute value. For example, if the input vector is [3/4 5/6 7/10], the output should be [45 50 42]; that is, the input vector should be multiplied by 60.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = integerize(x)\r\n  y = ceil(x);\r\nend","test_suite":"%%\r\nx = [3/4 5/6 7/10];\r\ny = integerize(x);\r\ny_correct = [45 50 42];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nx = [1.5 8.9 6.35 7.125];\r\ny = integerize(x);\r\ny_correct = [60 356 254 285];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nx = randi(80,[1 14])-40;\r\ny = integerize(x);\r\ny_correct = x;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nx = [4/5 -8/7 15/12 -13/49 83/56];\r\ny = integerize(x);\r\ny_correct = [1568 -2240 2450 -520 2905];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm = randi(6);\r\np = primes(100);\r\nk = randperm(length(p),2*m);\r\nn = p(k(1:m));\r\nd = p(k(m+1:end));\r\nf = prod(d);\r\nx = n./d;\r\ny = integerize(x);\r\ny_correct = x*f;\r\nassert(isequal(y,y_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2025-09-08T01:51:47.000Z","deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-09-08T01:51:22.000Z","updated_at":"2026-06-06T06:02:53.000Z","published_at":"2025-09-08T01:51:47.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes a vector of real numbers and multiplies them by a factor to produce the integers with the smallest absolute value. For example, if the input vector is [3/4 5/6 7/10], the output should be [45 50 42]; that is, the input vector should be multiplied by 60.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60975,"title":"Check Legendre's conjecture","description":"Context and problem statement\r\nLegendre's conjecture states for a given positive integer n, there always exists at least one prime number p between n² and (n+1²) :\r\n\r\n\r\n\r\nFor a given input n, List the prime number(s) p in the interval ]n^2; (n+1)^2[.\r\n\r\nExamples\r\n\r\nn = 1   =\u003e p = [2, 3];\r\nn = 2   =\u003e p = [5, 7];\r\nn = 10 =\u003e p = [101, 103, 107, 109, 113];\r\nn = 20 =\u003e p = [401, 409, 419, 421, 431, 433, 439];\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nProblem 60972. Check Bertrand-Chebyshev's theorem about prime numbers\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 697.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 348.733px; transform-origin: 408px 348.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 105px 8px; transform-origin: 105px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eContext and problem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.825px 8px; transform-origin: 174.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eLegendre's conjecture states for a given positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 144.308px 8px; transform-origin: 144.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethere always exists at least one prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3417px 8px; transform-origin: 30.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and (n+1²) :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"224\" height=\"19.5\" style=\"width: 224px; height: 19.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.0667px 8px; transform-origin: 54.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given input \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.675px 8px; transform-origin: 81.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, List the prime number(s) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 90px 8px; transform-origin: 90px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the interval ]n^2; (n+1)^2[.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 1   =\u0026gt; p = [2, 3];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 2   =\u0026gt; p = [5, 7];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 124.742px 8px; transform-origin: 124.742px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 10 =\u0026gt; p = [101, 103, 107, 109, 113];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 156.375px 8px; transform-origin: 156.375px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 20 =\u0026gt; p = [401, 409, 419, 421, 431, 433, 439];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60972\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60972. Check Bertrand-Chebyshev's theorem about prime numbers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = check_legendre_conjecture(n)\r\n  p = n;\r\nend","test_suite":"%%\r\nn = 1\r\np_correct = [2, 3];\r\nassert(isequal(check_legendre_conjecture(n),p_correct))\r\n\r\n%%\r\nn = 2\r\np_correct = [5, 7];\r\nassert(isequal(check_legendre_conjecture(n),p_correct))\r\n\r\n%%\r\nn = 10\r\np_correct = [101, 103, 107, 109, 113];\r\nassert(isequal(check_legendre_conjecture(n),p_correct))\r\n\r\n%%\r\nn = 20\r\np_correct = [401, 409, 419, 421, 431, 433, 439];\r\nassert(isequal(check_legendre_conjecture(n),p_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_legendre_conjecture.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:05:15.000Z","deleted_by":null,"deleted_at":null,"solvers_count":43,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-21T08:10:57.000Z","updated_at":"2026-07-20T06:46:32.000Z","published_at":"2025-07-21T08:28:05.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eContext and problem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLegendre's conjecture states for a given positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethere always exists at least one prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and (n+1²) :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall n \\\\in \\\\mathbb{N}^*, \\\\exists p \\\\in \\\\mathbb{P}, \\\\mathbf{n^2 \u0026lt; p \u0026lt; (n+1)^2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given input \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, List the prime number(s) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in the interval ]n^2; (n+1)^2[.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 1   =\u0026gt; p = [2, 3];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 2   =\u0026gt; p = [5, 7];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 10 =\u0026gt; p = [101, 103, 107, 109, 113];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 20 =\u0026gt; p = [401, 409, 419, 421, 431, 433, 439];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60972\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60972. Check Bertrand-Chebyshev's theorem about prime numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60970,"title":"Check p = 6n +/- 1, the generic formula for odd prime numbers greater than 3","description":"Problem statement\r\n\r\nFor all odd prime number p greater than 3, there exists a positive integer n, such that p = 6n +/- 1 :\r\n\r\n\r\n\r\nCheck this formula for some given odd primes in a vector by computing n for each p.\r\n\r\nExamples\r\n\r\np = 17 =\u003e n = 3;\r\n\r\np = 19 =\u003e n = 3;\r\n\r\n p = [5, 7, 11, 13, 17, 19] =\u003e n = [1, 1, 2, 2, 3, 3];\r\n\r\np = [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\r\n             =\u003e n = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12,  12, 13, 14, 15, 16];\r\n\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 919.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 459.733px; transform-origin: 408px 459.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.1333px 8px; transform-origin: 80.1333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor all odd prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.7333px 8px; transform-origin: 37.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003egreater than\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e 3\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.4583px 8px; transform-origin: 96.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, there exists a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3333px 8px; transform-origin: 30.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esuch that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep = 6n +/- 1 :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"231\" height=\"18\" style=\"width: 231px; height: 18px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 222.1px 8px; transform-origin: 222.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCheck this formula for some given odd primes in a vector by computing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.9417px 8px; transform-origin: 15.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eeach\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e p.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 17 =\u0026gt; n = 3;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 19 =\u0026gt; n = 3;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 148.817px 8px; transform-origin: 148.817px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 2, 3, 3];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 275.4px 8px; transform-origin: 275.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 266.008px 8px; transform-origin: 266.008px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e             =\u0026gt; n = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12,  12, 13, 14, 15, 16];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = check_odd_primes_gen_formula3(p)\r\n  n = p;\r\nend","test_suite":"%%\r\np = 17;\r\nn_correct = 3;\r\nassert(isequal(check_odd_primes_gen_formula3(p),n_correct))\r\n\r\n\r\n%%\r\np = 19;\r\nn_correct = 3;\r\nassert(isequal(check_odd_primes_gen_formula3(p),n_correct))\r\n\r\n\r\n%%\r\np = [5, 7, 11, 13, 17, 19];\r\nn_correct = [1, 1, 2, 2, 3, 3];\r\nassert(isequal(check_odd_primes_gen_formula3(p),n_correct))\r\n\r\n\r\n%%\r\np = [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\r\nn_correct = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12,  12, 13, 14, 15, 16];\r\nassert(isequal(check_odd_primes_gen_formula3(p),n_correct))\r\n\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_odd_primes_gen_formula3.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-20T05:22:24.000Z","deleted_by":null,"deleted_at":null,"solvers_count":42,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:25:50.000Z","updated_at":"2026-07-20T06:38:58.000Z","published_at":"2025-07-20T05:22:25.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor all odd prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003egreater than\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e 3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, there exists a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esuch that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 6n +/- 1 :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p \u0026gt; 3 \\\\Rightarrow \\\\exists n   \\\\in \\\\mathbb{N}, p = 6n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck this formula for some given odd primes in a vector by computing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eeach\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 17 =\u0026gt; n = 3;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 19 =\u0026gt; n = 3;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 2, 3, 3];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e             =\u0026gt; n = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12,  12, 13, 14, 15, 16];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60969,"title":"Check p = 4n +/- 1, the generic formula for odd prime numbers","description":"Problem statement\r\n\r\nFor all odd prime number p, there exists a positive integer n, such that p = 4n +/- 1 :\r\n\r\n\r\n\r\nCheck this formula for some given odd primes in a vector by computing n for each p.\r\n\r\nExamples\r\n\r\np = 17 =\u003e n = 4;\r\n\r\np = 19 =\u003e n = 5;\r\n\r\n p = [3, 5, 7, 11, 13, 17, 19] =\u003e n = [1, 1, 2, 3, 3, 4, 5];\r\n\r\np = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\r\n             =\u003e n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\r\n\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 919.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 459.733px; transform-origin: 408px 459.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.1333px 8px; transform-origin: 80.1333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor all odd prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.4583px 8px; transform-origin: 96.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, there exists a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3333px 8px; transform-origin: 30.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esuch that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep = 4n +/- 1 :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"231\" height=\"18\" style=\"width: 231px; height: 18px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 222.1px 8px; transform-origin: 222.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCheck this formula for some given odd primes in a vector by computing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.9417px 8px; transform-origin: 15.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eeach\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e p.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 17 =\u0026gt; n = 4;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 19 =\u0026gt; n = 5;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 164.367px 8px; transform-origin: 164.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 280.842px 8px; transform-origin: 280.842px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 287.408px 8px; transform-origin: 287.408px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e             =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = check_odd_primes_gen_formula(p)\r\n  n = p;\r\nend","test_suite":"%%\r\np = 17;\r\nn_correct = 4;\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = 19;\r\nn_correct = 5;\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = [3, 5, 7, 11, 13, 17, 19];\r\nn_correct = [1, 1, 2, 3, 3, 4, 5];\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\r\nn_correct = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_odd_primes_gen_formula.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-20T05:09:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":47,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:24:51.000Z","updated_at":"2026-07-20T06:37:47.000Z","published_at":"2025-07-20T05:09:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor all odd prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, there exists a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003esuch that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 4n +/- 1 :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p \u0026gt; 2 \\\\Rightarrow \\\\exists n   \\\\in \\\\mathbb{N}, p = 4n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck this formula for some given odd primes in a vector by computing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003eeach\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 17 =\u0026gt; n = 4;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 19 =\u0026gt; n = 5;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e             =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44230,"title":"I'm going to enjoy watching you calculate, Mr Anderson","description":"Smith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\r\n4+9+3+7+7+7+5=42\r\nThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\r\nSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 174px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 87px; transform-origin: 407px 87px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.5px 8px; transform-origin: 375.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSmith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e4+9+3+7+7+7+5=42\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 251px 8px; transform-origin: 251px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.5px 8px; transform-origin: 380.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = smith(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nassert(isequal(smith(4937775),1))\r\n%%\r\nassert(isequal(smith(1164),0))\r\n%%\r\nassert(isequal(smith(19683),1))\r\n%%\r\nassert(isequal(smith(11),0))  % Remember - Smith numbers are composite\r\n%%\r\nassert(isequal(smith(11^2),1))\r\n%%\r\nassert(isequal(smith(345741),1))\r\n%%\r\nassert(isequal(smith(19876),0))\r\n%%\r\nassert(isequal(smith(314159),0))\r\n%%\r\nassert(isequal(smith(612985),1))\r\n%%\r\nassert(isequal(smith(12379887),1))\r\n%%\r\nassert(isequal(smith(23456789),0))\r\n%%\r\nassert(isequal(smith(13),0))\r\n%%\r\nassert(isequal(smith(23),0))\r\n%%\r\ny=primes(randi(1e5));\r\nassert(isequal(smith(y(end)),0))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":1615,"edited_by":223089,"edited_at":"2023-01-07T08:26:27.000Z","deleted_by":null,"deleted_at":null,"solvers_count":49,"test_suite_updated_at":"2023-01-07T08:26:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2017-06-01T19:16:25.000Z","updated_at":"2026-03-16T15:29:16.000Z","published_at":"2017-06-01T19:16:25.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSmith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e4+9+3+7+7+7+5=42\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":53940,"title":"Easy Sequences 66: Fractional Part of Sum of Consecutive Powers over a Prime","description":"Given three postive integers, ,  and , create the function , defined as follows:\r\n            \r\nWhere  is the th prime and the symbol  is the  fractional part function of a positive real number, defined as follows: \r\n        If  then , where , is the 'floor' of .\r\nFor example for ,  and , we have:\r\n        \r\nTherefore, since :\r\n        \r\nPlease round-off the output to four decimal places, so in this case: .\r\nHINT: This is related to the Faulhaber's formula.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 355px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 177.5px; transform-origin: 407px 177.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 106px 8px; transform-origin: 106px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven three postive integers, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ee\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17px 8px; transform-origin: 17px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73px 8px; transform-origin: 73px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, create the function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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style=\"width: 85px; height: 18.5px;\" width=\"85\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62px 8px; transform-origin: 62px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, defined as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e            \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 181px; height: 45px;\" width=\"181\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.5px 8px; transform-origin: 77.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime and the symbol \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 18px; height: 18.5px;\" width=\"18\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the  \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://brilliant.org/wiki/factional-part-function/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003efractional part function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.5px 8px; transform-origin: 143.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a positive real number, defined as follows: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.5px 8px; transform-origin: 21.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 46px; height: 19px;\" width=\"46\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e then \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAAlCAYAAAAwTGn2AAAFVklEQVR4Xu1cOcsVSxD1+wEGLpGBgUsgCApuIBoYuEdi8DQTFNzAUFFf5gJqJriBoetDMxVNNNDEDTQyUQNBI1f8Ac9zpBuaYWbu6e66y9zbA8XcO0zXqao+1TNT3TNT08pWItCxCEx1zN5ibonAtELaQoLORSCFtLPg5SrIYsgnyC0Dr9dAx2qn7wX2Hw10FhX5EWBfrzfqY9WaHb3wYkhLB05A9kMuQe5BPkPeqta0nEfdiyCbnX4S94iRbgPzJlYFB5OnkBie5ARrHhp/6IWnGkNSPYQsh+zslQk5VqPtEsgTp4N4ZdTNDGhG80GTVsJTSbsXjl92I+yBjCCoTQeNp9o1aedJJDIMioSnkvYwDDvjLtlnDY1sUuWNf4QTNg0Ar0DUR0AikWHwJLxCWsOIj6EqiUSGfkt4hbSGER9DVRKJDP2W8AppDSM+hqokEhn6LeEV0hpGfAxVSSQy9FvCK6Q1jPgYqpJIZOi3hDcI0tKQLxBfb2UddjrkWYuz41Y9oM/c/EQMi+hzIO8g3ww73VqVRKIaUD9ZFPpHXb+DGNTZKuH1g7SssW6DLIDMd5bNdvvr2G90vznj1VQ+syQtdeVuvYJd1U/MfyHsPE6QcNsKeQA5BTnmjo16SU8ikfPlIvZMxpWQGZBXkBUQJuzdgAtr8btpwJLwVNLSIE7fxsyGvXQdRuNZa+WMGp3xRD6N38cb2ERH30A4pUfy52z/5zR2bW9jzznx2C0kKBOX0+Abghj4jo3VO6jzJRIFxjBJv7r/7N//IJzd/B747JO3zgcJTyEts4fB5bYQol7OPGlpPIlKApKk7Ej+52jTNkVLknNUbstMpfOoJ3d7DAUpkyqetIzfVcgeCEcfJsAuyElI221Srt257SUSBSAhaTnA0b9DkOeQCxCSt21GVcLrRdotALnmwLZjry6OqWYcRxd2VsxGHbyd4OXmKOQORE2YGJx+nusTlwuMSNSYpA/t8s8BObam3D9LJAqMIl/uu/8crLg1XU3rfJHwmkjLAN90Whnw2PUGofEcYWMIX3XGTyHzeIotOR2d0zZMXMbgHORKokJ/1Uls/rfZUog66HgciUSBUf7KQn9nOomxWcJTRtrzDjWGeP4emE1T7we7PtKGif8joQPDziYZlsX0fs25HHhiV8xJJAqw3uO3f2bZl5CkEl4v0tKelHva0PiUDCeu1T3tMKoHtD9M3JQOzOSoSXOJRA7Jr4Xl39QklfAU0oYdoATfP/nnjLI+ABZP18OqHvChg9WS1A40YV2mEolEDsMvJ+VfhSd1pkl4KmljliaGxreVN9riaVmnHUb1IEzcttJeJqf63lwikbOCr139436zvJfy0Czh9YO0/rKeM8JYkrbvPVsDED48pt4eDcPuKqZEItfIX9FSn2GoRsKzJm34xJzzpN910vpSl8XkyDDJK5EIBobVopgJqKQksSZtaHzqrUGYcaM+zVlHqGqNOqZOOUyCJt9jomF15i/l1mBoI61fd0ADcl6T6fJI69cdMAZdf6NYHWn9uoPX8DknSSU865HWaqToMmmtYjAKeiQSGRoq4RXSGkZ8DFVJJDL0W8IrpDWM+Biqkkhk6LeEp5LWT0nmVARifCvfPYiJVv/OlUhkCC/hqaQNvzCTUxVQ/Au/MLMODWIXeSgY5RwtAhKJNFXSWRKeSloiVr/ldQPHwtdoJKsaTvKvn5RveeVE0b5t+EBc1f4TBw5CUspb7G9WHKqbf9OjlZcxpPUA/pWKuThQvppoT5RR0kgS7W4w6BeOpy61zNKbQtpRCmqxZQIjUEg7gZ3edZcLabvegxNo/x/Jg0o14VJOFQAAAABJRU5ErkJggg==\" style=\"width: 86.5px; height: 18.5px;\" width=\"86.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25px 8px; transform-origin: 25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 23px; height: 18.5px;\" width=\"23\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50px 8px; transform-origin: 50px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is the 'floor' of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.5px 8px; transform-origin: 51.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 44px; height: 18px;\" width=\"44\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEgAAAAkCAYAAAAq23xmAAACq0lEQVRoQ+2YuUrEQBzG3QcQPDpBEK2sFLwaFRREsdVCwQdQCxtBQS1VUKwsFEHBTi1sxaMQPArxQCsL0TfwegL9PpiR2exMZncnS9zsBD6ySWYm+f/yv7KpMr+FEkh5PuEEPCCLh3hAHpBbEvEe5D0ojUA9jmrEmWfs323+VSoeNAgQ61BDAMgJjiehNxOoUgC0BOPnQjzlE9daTZCSDqgThl9Cm9CagFCNfR+0qHjUHX636SAmHdAtjKYYRsGNoF6gSnGhGfun4KAkA2qCsYdQB2RKxmr4jWPcVikBYmKu1RmtQJAhyFORAKJbNoobXOlitsjOqYC68OwZNmUbYiOYPC2MP8WeVeEVGtLFrQESXb48AoBRvhjatSds0YaiDRAbqwOIZXAU2hcGHmPfDy1D81kaLedkOVw7jCW5ymWBwNwNHE9As9Cqbt0wQHzj5xCzfDA+Pwznw56dCbHF0bgvzOdbj2KTVYy2GBO5CRAn30DsPOlB6kPN4HgFivptRmF0LmtIO7S5Ry5kAiRdj+PU/kDGLOH0QBl9Qy5PGONYRscjpKYN7ePoAMnJnCC9h9l+AWLeYbzuQNYPvRgBhN1aRgdts+ZPHSDVe9iCMwexcl1AZw5g/kMVIxwWC9pjhUPKOkBszVm1os4xcVexnOGYAP0I/zR+wOUZOnFXMb54m+ew+z5S7dN5kCzhBDRgCCkayzxk/B8lT4iFmsa0wZ6O9pg2FqBuKO3D1paDgv0P88i2UMaHXaGsc1xX5lQ2td+Gtepwng1jRsnXASJpmZy5HrP9A9QLtUNjQTd0NKCQ09WCY7uPNqWENYrDWJG9TgV0D10XERjCYD6ZslFRru/it/yU+jtt+xbLYf1kDvWALO/VA/KA3ELfe5D3IO9BbgS8B7nx8znIwu8XGuN3JcauM6EAAAAASUVORK5CYII=\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGIAAAAoCAYAAAAFZi8EAAAELUlEQVRoQ+1aS8tNURj+vh8gxEgGXy4DpUguEwwMECVFuYyUcp1JrkkGFCkj11ImbmVKpAyQkkvJFH0/wDU/gOf5rFevtdft7H3ss/Zp7Xo65zvr/j7rfd9nrf2NjpQnCwuMZjGLMomRQkQmm6AQUYjIxAKZTKN4RCEiEwukTWMaqk0GPqVVn6g1C5gB/ATexdoVjwhbiATsBI4AR4GrMYOi/BCwC5it6n6LtS9E+C1Lg5KAqabKngQiLqHOXuAyMA6MAVtVH4fx/ZxryEJE1SoMKfSC98BKY1jWihFB4lYBNLYORfSqh8BigJ4xF/hiD1uI8HsES5YDzxI8Qoy91mVk4xW3TT8LLaImfi5E9IeIBehmEvDc050QSo+gZ1SSfiGiP0SEe/mTJ+gRd413VOqHiKC7zQNmApRuohjW4ft809OLwC6ITa4L5amhKbYW5og5wCZXWPKFJiarzYCWYFQBt4AbAN2LSkLkGctOAJUEFJtdpJxGaPokafjAIP0ggkpqtYH3HBLyCD0JuhRjGzuUzj4oMlb8B8/41ZQFtPeGgsS+mxDByHHK2I3DfQR2+OwUImI3Gl0xE36NT1sRUK6dNeVefZy4YFc1unPT5wk6cOr2xI7rEMGIchDg51JAziEy5Hp8eWCPHyLiDipvMQ1ckksTcQb1jicurkvV6hBhr499XAAYUfjQM5YB/4TyEBESGh6hEb3BfjQR21BI4obt6QcRYpNXioxKKPcRkTIBho41AY9hEfvZburwtPkY6OXibNDEptghdY7MGfdN5cop3UfEaTQ4ZhpRHdnGo7T9bMqZP5ZYs2H5TYAxcp8hoFdVNSyqSZtGokwyEeJGvrCkE7mdfOS4TwI3AL7TZmwndV01udYna6rkXJdH6N3uUkMsfwnQ0C55KEm+qZLqqmrybTCqKCZqVwRx3jXpWOY6H8hVryv7S0x1KoOYC2RY3s8cIeImWb6KoWmX6YCO7XJn4jO0JonMi/xliGOu6FKi5vpTiZDN63sBJHbzynxXaNInZknUDEcHACbwUMiR3MJrj/PG8Dqf2MRm6AR/p8Q1X1SbiWF4v7UxpbIYWv7mRr1m/lhkwvhJfFYOctLAJkLiGMu5i98AP4Ax4ClA+RlSP75kJJ7SNG+0RZy85LHH+44f3gLXHXYgGRuBKaYRvX/c2LHnd9Z699a5PxIi7J0vrutTYW0ZONtxbI/Q1xp13lVIWLOJkFjrVAzZWqfFidnG/oqxeUlVd+dKCLK9SYhoehvaomnaHUoToRVC3VguIchuLyEv9gK+3dVnNJoQIVcScndUlwguTe6gdHiimuLje7mekUkGMxUSIW/kXDO4hx971f7yT1n81xKqLoY6qoc6fQ3GKgMYtU5CHsA0h3/IQkQmHBciChGZWCCTaRSPKERkYoFMplE8IhMifgP+q/MpP3bWugAAAABJRU5ErkJggg==\" style=\"width: 49px; height: 20px;\" width=\"49\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 35px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.5px; text-align: left; transform-origin: 384px 17.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; 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\" style=\"width: 430.5px; height: 35px;\" width=\"430.5\" height=\"35\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 210.5px 8px; transform-origin: 210.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ePlease round-off the output to four decimal places, so in this case: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 152px; height: 18.5px;\" width=\"152\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17px 8px; transform-origin: 17px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHINT:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.5px 8px; transform-origin: 68.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e This is related to the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Faulhaber%27s_formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eFaulhaber's formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = FPSP(x,e,n)\r\n    y = FPSP(x,e,n);\r\nend","test_suite":"%%\r\nx = 10; e = 2; n = 6;\r\ny_correct = 0.6154;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 100; e = 3; n = 50;\r\ny_correct = 0.6288;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 100; e = 4; n = 100;\r\ny_correct = 0.2495;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 250; e = 5; n = 200;\r\ny_correct = 0.1415;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 123; e = 6; n = 200;\r\ny_correct = 0.1292;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 10; e = 10; n = 300;\r\ny_correct = 0.7006;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 1234; e = 56; n = 789;\r\ny_correct = 0.8419;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 3000; e = 300; n = 3000;\r\ny_correct = 0.6441;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nxs = 20:2:2000;\r\nys = arrayfun(@(x) FPSP(x,x/2,x),xs);\r\nss = round([sum(ys) nnz(ys) mean(ys) mode(ys) median(ys) std(ys)],4); \r\nss_correct = [472.0982 numel(xs) 0.4764 0.2518 0.4704 0.2854];\r\nassert(isequal(ss,ss_correct))\r\n%%\r\nfiletext = fileread('FPSP.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'BigInteger') || contains(filetext, 'java'); \r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":"2022-01-20T12:19:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-01-19T18:18:26.000Z","updated_at":"2026-05-24T23:36:48.000Z","published_at":"2022-01-20T10:08:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven three postive integers, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, create the function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eFPSP(x,y,n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, defined as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e            \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eFPSP(x,e,n)=\\\\left\\\\{ \\\\frac{1}{p_{n}} \\\\cdot \\\\sum^{x}_{i=1} i^{e}\\\\right\\\\}  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_{n}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime and the symbol \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\{\\\\}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://brilliant.org/wiki/factional-part-function/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003efractional part function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e of a positive real number, defined as follows: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er \\\\in \\\\mathbb{R}^{+}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\{r\\\\}=r-\\\\lfloor r \\\\rfloor\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lfloor r \\\\rfloor\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is the 'floor' of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee=2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en=6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sum^{10}_{i=1} i^{2} = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTherefore, since \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_6=13\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eFPSP(10,2,6)=\\\\left\\\\{ \\\\frac{385}{13} \\\\right\\\\}  = \\\\{29.615384561538...\\\\}=0.61538461538...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease round-off the output to four decimal places, so in this case: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eFPSP(10,2,6)=0.6154\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHINT:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e This is related to the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Faulhaber%27s_formula\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eFaulhaber's formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":61052,"title":"Compute the Dedekind psi function","description":"Write a function to compute the Dedekind psi function. The values for n = 1, 2, 3, 4, 5, and 6 are 1, 3, 4, 6, 6, and 12. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 21px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 10.5px; transform-origin: 408px 10.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/DedekindFunction.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eDedekind psi function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The values for n = 1, 2, 3, 4, 5, and 6 are 1, 3, 4, 6, 6, and 12. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = DedekindPsi(n)\r\n  y = psi(n);\r\nend","test_suite":"%%\r\nn = 1;\r\ny = DedekindPsi(n);\r\ny_correct = 1;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 8;\r\ny = DedekindPsi(n);\r\ny_correct = 12;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 49;\r\ny = DedekindPsi(n);\r\ny_correct = 56;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 312;\r\ny = DedekindPsi(n);\r\ny_correct = 672;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 1441;\r\ny = DedekindPsi(n);\r\ny_correct = 1584;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 8028;\r\ny = DedekindPsi(n);\r\ny_correct = 16128;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 14035;\r\ny = DedekindPsi(n);\r\ny_correct = 19296;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 85432;\r\ny = DedekindPsi(n);\r\ny_correct = 131040;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 6645344;\r\ny = DedekindPsi(n);\r\ny_correct = 10402560;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 5533221188;\r\ny = DedekindPsi(n);\r\ny_correct = 9083498112;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\np = primes(1e8);\r\nq = p(randi(length(p)));\r\ny = DedekindPsi(q);\r\nassert(isequal(y,q+1))\r\n\r\n%%\r\np = primes(200);\r\nq = p(randi(length(p)));\r\nn = randi(5);\r\nx = q^n;\r\ny = DedekindPsi(x);\r\nassert(isequal(y,x+x/q))\r\n\r\n%%\r\np = primes(200);\r\nk = unique(randi(length(p),[1 randi(5)]));\r\nq = p(k);\r\nx = prod(q);\r\ny = DedekindPsi(x);\r\nassert(isequal(y,prod(q+1)))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2025-10-25T15:14:46.000Z","deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-10-25T15:09:34.000Z","updated_at":"2026-06-06T15:56:45.000Z","published_at":"2025-10-25T15:14:46.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/DedekindFunction.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDedekind psi function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. The values for n = 1, 2, 3, 4, 5, and 6 are 1, 3, 4, 6, 6, and 12. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1927,"title":"Find nth maximum","description":"Find nth maximum in a vector of integer numbers. Return NaN if no such number exists.\r\n\r\n x = [2 6 4 9 -10 3 1 5 -10];\r\n\r\nSo \r\n\r\n* n_max(x,3) would find 3rd maximum value in x which is 5\r\n* n_max(x,8) would find 8th maximum value in x which is -10\r\n* n_max(x,9) would find 9th maximum value in x which is -10\r\n* n_max(x,12) does not exist so return NaN\r\n\r\n","description_html":"\u003cp\u003eFind nth maximum in a vector of integer numbers. Return NaN if no such number exists.\u003c/p\u003e\u003cpre\u003e x = [2 6 4 9 -10 3 1 5 -10];\u003c/pre\u003e\u003cp\u003eSo\u003c/p\u003e\u003cul\u003e\u003cli\u003en_max(x,3) would find 3rd maximum value in x which is 5\u003c/li\u003e\u003cli\u003en_max(x,8) would find 8th maximum value in x which is -10\u003c/li\u003e\u003cli\u003en_max(x,9) would find 9th maximum value in x which is -10\u003c/li\u003e\u003cli\u003en_max(x,12) does not exist so return NaN\u003c/li\u003e\u003c/ul\u003e","function_template":"function y = n_max(x,n)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [2 6 4 9 -10 3 1 5 -10];\r\ny_correct = 5;\r\nassert(isequaln(n_max(x,3),y_correct))\r\n\r\n%%\r\nx = [2 6 4 9 -10 3 1 5 -10];\r\ny_correct = NaN;\r\nassert(isequaln(n_max(x,13),y_correct))\r\n\r\n%%\r\nx = [0 0 0 -2 0 -6];\r\ny_correct = -2;\r\nassert(isequaln(n_max(x,5),y_correct))\r\n\r\n%%\r\nx = [1 1 1 1 1 1];\r\ny_correct = 1;\r\nassert(isequaln(n_max(x,4),y_correct))\r\n\r\n%%\r\nx = [1 1 1 1 1 1];\r\ny_correct = NaN;\r\nassert(isequaln(n_max(x,14),y_correct))","published":true,"deleted":false,"likes_count":6,"comments_count":4,"created_by":18172,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":942,"test_suite_updated_at":"2013-10-15T19:44:56.000Z","rescore_all_solutions":false,"group_id":13,"created_at":"2013-10-10T06:23:03.000Z","updated_at":"2026-07-19T12:00:35.000Z","published_at":"2013-10-10T06:28:51.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind nth maximum in a vector of integer numbers. Return NaN if no such number exists.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [2 6 4 9 -10 3 1 5 -10];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSo\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en_max(x,3) would find 3rd maximum value in x which is 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en_max(x,8) would find 8th maximum value in x which is -10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en_max(x,9) would find 9th maximum value in x which is -10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en_max(x,12) does not exist so return NaN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60977,"title":"Linear primes p' = mp + n","description":"Problem statement\r\n\r\nList the prime numbers of the form p' = mp + n for a given (m,n) couple and such that p' \u003c k, k positive integer.\r\n\r\nExamples \r\n\r\n(m,n) = (3,2) and k = 50   =\u003e p = [11, 17, 23, 41];\r\n(m,n) = (4,3) and k = 50   =\u003e p = [11, 23, 31, 47];\r\n(m,n) = (5,4) and k = 100 =\u003e p = [19, 29, 59, 89];\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 536.033px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 268.017px; transform-origin: 408px 268.017px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.508px 8px; transform-origin: 108.508px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eList the prime numbers of the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.7917px 8px; transform-origin: 34.7917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep' = mp + n\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5583px 8px; transform-origin: 36.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for a given \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.3333px 8px; transform-origin: 16.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(m,n)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.4583px 8px; transform-origin: 68.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e couple and such that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.025px 8px; transform-origin: 26.025px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep' \u0026lt; k, k \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.625px 8px; transform-origin: 48.625px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003epositive integer.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 149.425px 8px; transform-origin: 149.425px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(m,n) = (3,2) and k = 50   =\u0026gt; p = [11, 17, 23, 41];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 149.425px 8px; transform-origin: 149.425px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(m,n) = (4,3) and k = 50   =\u0026gt; p = [11, 23, 31, 47];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 149.95px 8px; transform-origin: 149.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(m,n) = (5,4) and k = 100 =\u0026gt; p = [19, 29, 59, 89];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = list_the_linear_primes(m,n,k)\r\n  p = m;\r\nend","test_suite":"%%\r\nm = 3;\r\nn = 2;\r\nk = 50;\r\np_correct = [11, 17, 23, 41];\r\nassert(isequal(list_the_linear_primes(m,n,k),p_correct))\r\n\r\n%%\r\nm = 4;\r\nn = 3;\r\nk = 50;\r\np_correct = [11, 23, 31, 47];\r\nassert(isequal(list_the_linear_primes(m,n,k),p_correct))\r\n\r\n%%\r\nm = 5;\r\nn = 4;\r\nk = 100;\r\np_correct = [19, 29, 59, 89];\r\nassert(isequal(list_the_linear_primes(m,n,k),p_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('list_the_linear_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-21T10:46:17.000Z","deleted_by":null,"deleted_at":null,"solvers_count":37,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-21T10:27:56.000Z","updated_at":"2026-07-20T06:35:34.000Z","published_at":"2025-07-21T10:46:17.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eList the prime numbers of the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep' = mp + n\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for a given \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(m,n)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e couple and such that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep' \u0026lt; k, k \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003epositive integer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(m,n) = (3,2) and k = 50   =\u0026gt; p = [11, 17, 23, 41];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(m,n) = (4,3) and k = 50   =\u0026gt; p = [11, 23, 31, 47];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(m,n) = (5,4) and k = 100 =\u0026gt; p = [19, 29, 59, 89];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52921,"title":"Easy Sequences 40: Quadratic Congruence","description":"Quadratic Congruence is a modular equation of the form: . \r\nIn this exercise you will be given a vector containing the coefficients of a quadratic polynomial (), and a modulus base (). Using these data, create a function that outputs the pair (), which are the 'primitive' solutions to the quadratic congruence.\r\nFor example consider the congruence: ,  the solution is , since:\r\n                                    , and\r\n                                    .\r\nNOTE: A primitive modulus to base , can only have values from  to . This is a simplified problem, in which the quadratic polynomials given in the test suite, are all factorable, and the modulus base are all odd primes.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 235px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 117.5px; transform-origin: 407px 117.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297px 8px; transform-origin: 297px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn this exercise you will be given a vector containing the coefficients of a quadratic polynomial (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 79px; height: 18.5px;\" width=\"79\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.5px 8px; transform-origin: 26.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e), and a modulus base (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.5px 8px; transform-origin: 6.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e). \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 200.5px 8px; transform-origin: 200.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eUsing these data, create a function that outputs the pair (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 75px; height: 20px;\" width=\"75\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 56px 8px; transform-origin: 56px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e), which are the 'primitive' solutions to the quadratic congruence.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 123.5px 8px; transform-origin: 123.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example consider the congruence: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 157.5px; height: 19.5px;\" width=\"157.5\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.5px 8px; transform-origin: 51.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e,  the solution is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 35px; height: 18.5px;\" width=\"35\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.5px 8px; transform-origin: 22.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, since:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 292px; height: 19.5px;\" width=\"292\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 292px; height: 19.5px;\" width=\"292\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNOTE: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.5px 8px; transform-origin: 88.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA primitive modulus to base \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.5px 8px; transform-origin: 88.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, can only have values from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003e0\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10px 8px; transform-origin: 10px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE0AAAAkCAYAAADM8rciAAABnElEQVRoge2YYZGEMAyFnwccYGANoAAFOMABDrBwGpCAByygAQt3P9oMuWyPG0hnoTPvm+mfvqXtvk2TsAAhhBBCiIfX3QcoiQ7ACmC6+yAl0CKY9R0HTfuHEUAfB027AE27AE27QHbTKgCNmavjXJ34/JH2VNymNQAGADOALS42RK2KC8smG3ZzKoTkKtqCcozLEmnWnCbOLQjGDEprlTYntBLIdj3FtA27KX3UWrVRfaCV0mFnM02avgkhgnqlSTStCJE3Kk2u6HZirxdCNHtGdebLGbKYVquFZgBfRp+x563FaEvU7DNHyHqe4YnqLKZ1aqEV77+iPqxO9pWa707sNyIY5xmeopPFNF0EbDLXOWs80EqpnEAm06TVWBOabitsBIqWeu7JuE3T0dIndMlZqQ1EsxH4dNym6UiyV0wXCJuzdD7TvZunqn0Kt2nSaswJTf+NYs1ojDbhXDG4E5dpOpJSV1MKhG0zgN9vAusfzz8Re+7TBazFXsJT10q0VAS9ojbh/QX/iXTYG3c7BpTzNkMIIYQQQsh1fgCYxc53qqxS4gAAAABJRU5ErkJggg==\" style=\"width: 38.5px; height: 18px;\" width=\"38.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132px 8px; transform-origin: 132px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. This is a simplified problem, in which the quadratic polynomials given in the test suite, are all factorable, and the modulus base are all odd primes.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function xs = quadCong(cs,m)\r\n    xs = cs;\r\nend","test_suite":"%%\r\ncs = [1,5,6]; m = 3;\r\nxs_correct = [0,1];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [1,5,6]; m = 7;\r\nxs_correct = [4,5];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [6,-7,-5]; m = 11;\r\nxs_correct = [5,9];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [35,76,33]; ms = setdiff(primes(10000),primes(100));\r\nys = arrayfun(@(m) sum(quadCong(cs,m)),ms);\r\nss = [sum(ys) floor([mean(ys) mode(ys) median(ys) std(ys)])];\r\nss_correct = [5752615 4777 57 4151 3465];\r\nassert(isequal(ss,ss_correct))\r\n%%\r\ncs = [10,-131,391]; m = 6373;\r\nxs_correct = [3195,5103];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [253,-1420,672]; m = 30103;\r\nxs_correct = [3927,8215];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [528,26711,283500]; m = 198733;\r\nxs_correct = [37227,174629];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [4545,13092,-69741]; m = 1000121;\r\nxs_correct = [821876,933449];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\nfiletext = fileread('quadCong.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'BigInteger') || contains(filetext, 'java'); \r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":19,"test_suite_updated_at":"2021-10-17T05:04:28.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-10-12T12:11:53.000Z","updated_at":"2026-05-24T20:40:09.000Z","published_at":"2021-10-16T17:40:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eQuadratic Congruence is a modular equation of the form: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eax^{2}+bx+c\\\\equiv 0\\\\  (mod\\\\  m)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn this exercise you will be given a vector containing the coefficients of a quadratic polynomial (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ecs =\\\\left[ a,b,c\\\\right]  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e), and a modulus base (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e). \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eUsing these data, create a function that outputs the pair (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003exs=[x_{1},x_{2}]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e), which are the 'primitive' solutions to the quadratic congruence.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example consider the congruence: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^{2}+5x+6\\\\equiv 0\\\\  (mod\\\\  7)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,  the solution is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e[4,5]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, since:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e4^{2}+5\\\\cdot 4+6= 16+20+6= 42\\\\equiv 0\\\\  (mod\\\\  7)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e5^{2}+5\\\\cdot 5+6= 25+25+6= 56\\\\equiv 0\\\\  (mod\\\\  7)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNOTE: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eA primitive modulus to base \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, can only have values from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em-1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. This is a simplified problem, in which the quadratic polynomials given in the test suite, are all factorable, and the modulus base are all odd primes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44068,"title":"The number of trailing zero digit of a factorial","description":"For given positive integer n, take factorial of that number. How many trailing zeros does it have?\r\n\r\nExample: factorial(11) = 39916800\r\n\r\nIts last zero-digit count is 2.\r\n\r\nOptional: Can you make an efficient algorithm for a very large n?","description_html":"\u003cp\u003eFor given positive integer n, take factorial of that number. How many trailing zeros does it have?\u003c/p\u003e\u003cp\u003eExample: factorial(11) = 39916800\u003c/p\u003e\u003cp\u003eIts last zero-digit count is 2.\u003c/p\u003e\u003cp\u003eOptional: Can you make an efficient algorithm for a very large n?\u003c/p\u003e","function_template":"function ct = powerTenInFactorial(n)\r\n  ct = 0;\r\nend","test_suite":"%%\r\nn = 1;\r\nct_correct = 0;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))\r\n\r\n%%\r\nn = 9;\r\nct_correct = 1;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))\r\n\r\n%%\r\nn = 27;\r\nct_correct = 6;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))\r\n\r\n%%\r\nn = 626;\r\nct_correct = 156;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))\r\n\r\n%%\r\nn = 620;\r\nct_correct = 152;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":3,"created_by":115733,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":66,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":673,"created_at":"2017-02-14T00:24:29.000Z","updated_at":"2026-03-20T13:50:01.000Z","published_at":"2017-02-14T00:24:29.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor given positive integer n, take factorial of that number. How many trailing zeros does it have?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample: factorial(11) = 39916800\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIts last zero-digit count is 2.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOptional: Can you make an efficient algorithm for a very large n?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1771,"title":"Polygonal numbers","description":"The task of \u003chttp://www.mathworks.co.uk/matlabcentral/cody/problems/5 Problem 5\u003e is to calculate triangular numbers. By playing with dots we can produce also square numbers like:\r\n\r\n                                 * * * *\r\n                     * * *       * * * *\r\n           * *       * * *       * * * *\r\n1: *    4: * *    9: * * *   16: * * * *\r\n\r\nor hexagonal numbers:\r\n\r\n                                          * * * *\r\n                                         *       *\r\n                         * * *          * * * *   *\r\n                        *     *        * *     *   *\r\n            * *        * * *   *        * * *   * * \r\n           *   *        *   * *          *   * * *\r\n1: *    6:  * *    15:   * * *       28:  * * * *\r\n\r\n\u0026nbsp\r\n\r\nAccording to those rules we can create \u003chttp://en.wikipedia.org/wiki/Polygonal_number polygonal numbers\u003e for all regular polygons.\r\n\r\n\u0026nbsp\r\n\r\nYour task: given _S_ and _N_ *calculate _N_-th _S_-gonal numbers* _P(S,N)_ \r\n\r\n\u0026nbsp\r\n\r\nExamples:\r\n\r\n# _P(4, 3)_ returns _[9]_    because 3-rd square number is 9,\r\n# _P(3, 1:5)_ returns _[1, 3, 6, 10, 15]_   first 5 triangular numbers,\r\n# _P(3:6, 4)_ returns _[10, 16, 22, 28]_   4-th triangular, square, pentagonal and hexagonal numbers,\r\n# _P([3, 4], [1; 2])_ returns _[1, 1; 3, 4]_.\r\n\r\nsee the test suite for more hints","description_html":"\u003cp\u003eThe task of \u003ca href = \"http://www.mathworks.co.uk/matlabcentral/cody/problems/5\"\u003eProblem 5\u003c/a\u003e is to calculate triangular numbers. By playing with dots we can produce also square numbers like:\u003c/p\u003e\u003cpre\u003e                                 * * * *\r\n                     * * *       * * * *\r\n           * *       * * *       * * * *\r\n1: *    4: * *    9: * * *   16: * * * *\u003c/pre\u003e\u003cp\u003eor hexagonal numbers:\u003c/p\u003e\u003cpre\u003e                                          * * * *\r\n                                         *       *\r\n                         * * *          * * * *   *\r\n                        *     *        * *     *   *\r\n            * *        * * *   *        * * *   * * \r\n           *   *        *   * *          *   * * *\r\n1: *    6:  * *    15:   * * *       28:  * * * *\u003c/pre\u003e\u003cp\u003e\u0026nbsp\u003c/p\u003e\u003cp\u003eAccording to those rules we can create \u003ca href = \"http://en.wikipedia.org/wiki/Polygonal_number\"\u003epolygonal numbers\u003c/a\u003e for all regular polygons.\u003c/p\u003e\u003cp\u003e\u0026nbsp\u003c/p\u003e\u003cp\u003eYour task: given \u003ci\u003eS\u003c/i\u003e and \u003ci\u003eN\u003c/i\u003e \u003cb\u003ecalculate \u003ci\u003eN\u003c/i\u003e-th \u003ci\u003eS\u003c/i\u003e-gonal numbers\u003c/b\u003e \u003ci\u003eP(S,N)\u003c/i\u003e\u003c/p\u003e\u003cp\u003e\u0026nbsp\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003col\u003e\u003cli\u003e\u003ci\u003eP(4, 3)\u003c/i\u003e returns \u003ci\u003e[9]\u003c/i\u003e    because 3-rd square number is 9,\u003c/li\u003e\u003cli\u003e\u003ci\u003eP(3, 1:5)\u003c/i\u003e returns \u003ci\u003e[1, 3, 6, 10, 15]\u003c/i\u003e   first 5 triangular numbers,\u003c/li\u003e\u003cli\u003e\u003ci\u003eP(3:6, 4)\u003c/i\u003e returns \u003ci\u003e[10, 16, 22, 28]\u003c/i\u003e   4-th triangular, square, pentagonal and hexagonal numbers,\u003c/li\u003e\u003cli\u003e\u003ci\u003eP([3, 4], [1; 2])\u003c/i\u003e returns \u003ci\u003e[1, 1; 3, 4]\u003c/i\u003e.\u003c/li\u003e\u003c/ol\u003e\u003cp\u003esee the test suite for more hints\u003c/p\u003e","function_template":"function y = P(s,n)\r\n  y = s*n;\r\nend","test_suite":"%%\r\nassert(P(4,3)==9)\r\n%%\r\ns=3;\r\nn=1:5;\r\ny_correct=[1 3 6 10 15];\r\nassert(isequal(P(s,n),y_correct))\r\n%%\r\ns=3:6;\r\nn=4;\r\nY_correct=[10 16 22 28];\r\nassert(isequal(P(s,n),Y_correct))\r\n%%\r\ns=randi([3 1000],1,10);\r\nn=1;\r\ny_correct=ones(1,10);\r\nassert(isequal(P(s,n),y_correct))\r\n%%\r\ns=randi(1000,1,10)+2;\r\nn=5;\r\ny_correct=10*s-15;\r\nassert(isequal(P(s,n),y_correct))\r\n%%\r\ns=[5 6 7]';\r\nn=[2 8 9];\r\ny_correct=[5 92 117;6 120 153;7 148 189];\r\nassert(isequal(P(s,n),y_correct))\r\n%%\r\n% for M.Z.\r\ns=[5 6 7]';\r\nn=[2 8 9];\r\ny_correct=[5 92 117;6 120 153;7 148 189];\r\nassert(isequal(P(flipud(s),n),flipud(y_correct)))\r\n%%\r\ns=randi([3,1000]);\r\nn=randi([3,1000]);\r\nassert(P(s+1,n+1)-P(s,n+1)==P(3,n))\r\n","published":true,"deleted":false,"likes_count":5,"comments_count":4,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":140,"test_suite_updated_at":"2016-10-15T05:56:15.000Z","rescore_all_solutions":true,"group_id":8,"created_at":"2013-08-02T13:16:29.000Z","updated_at":"2026-07-15T12:56:22.000Z","published_at":"2013-08-02T23:34:24.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe task of\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.co.uk/matlabcentral/cody/problems/5\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is to calculate triangular numbers. By playing with dots we can produce also square numbers like:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[                                 * * * *\\n                     * * *       * * * *\\n           * *       * * *       * * * *\\n1: *    4: * *    9: * * *   16: * * * *]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eor hexagonal numbers:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[                                          * * * *\\n                                         *       *\\n                         * * *          * * * *   *\\n                        *     *        * *     *   *\\n            * *        * * *   *        * * *   * * \\n           *   *        *   * *          *   * * *\\n1: *    6:  * *    15:   * * *       28:  * * * *]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026amp;nbsp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAccording to those rules we can create\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Polygonal_number\\\"\u003e\u003cw:r\u003e\u003cw:t\u003epolygonal numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for all regular polygons.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026amp;nbsp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task: given\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ecalculate\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e-th\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e-gonal numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP(S,N)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026amp;nbsp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP(4, 3)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[9]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e because 3-rd square number is 9,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP(3, 1:5)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[1, 3, 6, 10, 15]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e first 5 triangular numbers,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP(3:6, 4)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[10, 16, 22, 28]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e 4-th triangular, square, pentagonal and hexagonal numbers,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP([3, 4], [1; 2])\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[1, 1; 3, 4]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003esee the test suite for more hints\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46054,"title":"Count trailing zeros in a primorial","description":"\u003chttps://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial Cody Problem 44068\u003e asked us to count the  trailing zeros in a factorial. This problem deals with the  _\u003chttps://mathworld.wolfram.com/Primorial.html primorial\u003e_. If p_n is the nth prime number, then the primorial p_n# is the product of the prime numbers up to and including p_n. For example, if n = 5, then p_5# = 2*3*5*7*11 = 2310, which has one trailing zero. \r\n\r\nCount the trailing zeros in the primorial p_n#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.* \r\n\r\nPlease make your code general because I might add more tests later. \r\n\r\n*;-)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 205px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 102.5px; transform-origin: 407px 102.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.8833px; text-align: left; transform-origin: 384px 31.8833px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 44068\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 238.833px 7.91667px; transform-origin: 238.833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asked us to count the trailing zeros in a factorial. This problem deals with the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/Primorial.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eprimorial\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 7.91667px; transform-origin: 9.71667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6167px 7.91667px; transform-origin: 20.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.28333px 7.91667px; transform-origin: 7.28333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime number, then the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 178.55px 7.91667px; transform-origin: 178.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# is the product of the prime numbers up to and including \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 52.1167px 7.91667px; transform-origin: 52.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.85px 7.91667px; transform-origin: 29.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 5, then \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_5\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.8667px 7.91667px; transform-origin: 11.8667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2*3*5*7*11\" style=\"width: 92px; height: 18px;\" width=\"92\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 112.233px 7.91667px; transform-origin: 112.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 2310, which has one trailing zero.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.1333px; text-align: left; transform-origin: 384px 21.1333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 121.767px 7.91667px; transform-origin: 121.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCount the trailing zeros in the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.467px 7.91667px; transform-origin: 254.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.1px 7.91667px; transform-origin: 215.1px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ePlease make your code general because I might add more tests later.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problems 46057 \"Find the last non-zero digit in a primorial\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.91667px; transform-origin: 15.5667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46060 \"Identify primorial primes\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.33333px 7.91667px; transform-origin: 9.33333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e*;-)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = primorialTrailingZeros(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = 0;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 2;\r\ny_correct = 0;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 5;\r\ny_correct = 1;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 10;\r\ny_correct = sin((n+3)*pi/2);\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 100;\r\ny_correct = sum(1./2.^(1:n));\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 1e4;\r\ny_correct = besselj(mod(n,10),prod(num2str(n)'-'0'));\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 1e6;\r\nf = [factor(log10(n)) 3 7 7 89];\r\ny_correct = length(primes(n))/prod(f);\r\nassert(isequal(primorialTrailingZeros(n),y_correct))","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":84,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-26T15:54:06.000Z","updated_at":"2026-05-22T14:07:45.000Z","published_at":"2020-07-26T16:20:27.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 44068\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asked us to count the trailing zeros in a factorial. This problem deals with the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/Primorial.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprimorial\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime number, then the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# is the product of the prime numbers up to and including \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 5, then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2*3*5*7*11\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\cdot3\\\\cdot5\\\\cdot7\\\\cdot11\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 2310, which has one trailing zero.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCount the trailing zeros in the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease make your code general because I might add more tests later.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems 46057 \\\"Find the last non-zero digit in a primorial\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46060 \\\"Identify primorial primes\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e*;-)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60943,"title":"Ulam primes second formula","description":"Historical context\r\n\r\n\r\nThe polish-american mathematician Stanislaw Ulam found the formula  to give some prime numbers not given neither by Euler's formula neither by its own first formula. \r\n\r\n\r\nProblem statement\r\n\r\nFor a given integer m, m \u003e= 59, list the Ulam prime numbers less or equal to m in a vector up. Also compute their frequency / ratio f compare to all the primes in the same range. If m is not a positive integer greater or equal to 59, your function should return up = [] and f = 0.\r\n\r\nExamples (check the tests for more)\r\n\r\nIf m = 59 then up = 59 and f = 1 since 59 = P(0) and it is the only prime number m greater or equal and less or equal to 59;\r\nIf m = 200 then up = [59, 67, 83, 107, 139, 179] and f = 6/30 since 59 = P(0), 67 = P(1), 83 = P(2), 107 = P(3), 139 = P(4), 179 = P(5), and there are in total 30 prime numbers in the range [59; 200] ; \r\nIf m = 23 then up = [] and f = 0;\r\n\r\n\r\nTip \r\n\r\n here.\r\n\r\n\r\nFobidden functions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\n\r\n\r\nSee also\r\nUlam primes first formula\r\nEuler primes\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1069.47px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 534.733px; transform-origin: 408px 534.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 219.4px 8px; transform-origin: 219.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe polish-american mathematician Stanislaw Ulam found the formula \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"140\" height=\"19.5\" style=\"width: 140px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.1833px 8px; transform-origin: 92.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to give some prime numbers not given neither by Euler's formula neither by its own first formula. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.075px 8px; transform-origin: 26.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u0026gt;= 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 138.475px 8px; transform-origin: 138.475px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Ulam prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0583px 8px; transform-origin: 40.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector u\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.3833px 8px; transform-origin: 96.3833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 294.442px 8px; transform-origin: 294.442px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.35px 8px; transform-origin: 63.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.075px 8px; transform-origin: 25.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e up = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3333px 8px; transform-origin: 79.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests for more)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 51.0833px; transform-origin: 392px 51.0833px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6.21667px 8px; transform-origin: 6.21667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e u\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.0333px 8px; transform-origin: 20.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.1417px 8px; transform-origin: 16.1417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.0667px 8px; transform-origin: 26.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.2px 8px; transform-origin: 21.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e = P(0)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 99.575px 8px; transform-origin: 99.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and it is the only prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.533px 8px; transform-origin: 108.533px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e greater or equal and less or equal to 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2.33333px 8px; transform-origin: 2.33333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 200\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6.21667px 8px; transform-origin: 6.21667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e u\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 94.7083px 8px; transform-origin: 94.7083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [59, 67, 83, 107, 139, 179]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 6/30 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.0667px 8px; transform-origin: 26.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 147.225px 8px; transform-origin: 147.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e = P(0), 67 = P(1), 83 = P(2), 107 = P(3), 139 = P(4), 179 = P(5), \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 73.9083px 8px; transform-origin: 73.9083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand there are in total 30\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.9917px 8px; transform-origin: 34.9917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [59; 200] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 23 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.45px 8px; transform-origin: 19.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen u\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.9167px 8px; transform-origin: 16.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630/problems/60944\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eUlam primes first formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60942-euler-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eEuler primes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [up,f] = ulam_primes2(m)\r\n  up = m;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 59;\r\nup_correct = 59;\r\nf_correct = 1;\r\n[up,f] = ulam_primes2(m);\r\nassert(isequal([up,f],[up_correct,f_correct]));\r\n\r\n%%\r\nm = 200;\r\nup_correct = [59, 67, 83, 107, 139, 179];\r\nf_correct = 1/5;\r\n[up,f] = ulam_primes2(m);\r\nassert(isequal([up,f],[up_correct,f_correct]));\r\n\r\n%%\r\nm = 500;\r\nup_correct = [59, 67, 83, 107, 139, 179, 227, 283, 347, 419, 499];\r\nf_correct = 11/79;\r\n[up,f] = ulam_primes2(m);\r\nassert(isequal([up,f],[up_correct,f_correct]));\r\n\r\n%%\r\nm = 23;\r\nup_correct = [];\r\nf_correct = 0;\r\n[up,f] = ulam_primes2(m);\r\nassert(isequal([up,f],[up_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('ulam_primes2.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:08:26.000Z","deleted_by":null,"deleted_at":null,"solvers_count":22,"test_suite_updated_at":"2025-07-09T05:56:45.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-26T19:08:27.000Z","updated_at":"2026-06-09T14:26:47.000Z","published_at":"2025-06-26T19:44:55.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe polish-american mathematician Stanislaw Ulam found the formula \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{P(n) = 4n^2 + 4n + 59}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to give some prime numbers not given neither by Euler's formula neither by its own first formula. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u0026gt;= 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Ulam prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector u\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e up = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e u\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e = P(0)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and it is the only prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e greater or equal and less or equal to 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 200\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e u\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [59, 67, 83, 107, 139, 179]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 6/30 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e = P(0), 67 = P(1), 83 = P(2), 107 = P(3), 139 = P(4), 179 = P(5), \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand there are in total 30\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [59; 200] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 23 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen u\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":49582,"title":"Determine whether a square can be drawn on a grid of dots","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 444.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 222.233px; transform-origin: 407px 222.233px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.358px 7.79167px; transform-origin: 382.358px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSuppose you have a regular grid of dots, each 1 unit apart. By connecting four dots arranged in a square of side 1, you can draw a square of area 1. Similarly, you can draw squares of area 4, 9, 16, 25, etc. As shown below, by connecting dots with diagonal lines, you can draw squares of area 2, 5, 10, 8, 13, and 17. Which squares can you draw by connecting dots on the grid with straight lines? \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 301.708px 7.79167px; transform-origin: 301.708px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes an area and determines whether the square of that area can be drawn.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 7.79167px; transform-origin: 0px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 291.467px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 145.733px; text-align: left; transform-origin: 384px 145.733px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://www.mathsisfun.com/activity/images/squares8.gif\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = squareOnGrid(A)\r\n  tf = false;\r\nend","test_suite":"%%\r\nassert(squareOnGrid(1))\r\n\r\n%%\r\nassert(squareOnGrid(2))\r\n\r\n%%\r\nassert(~squareOnGrid(3))\r\n\r\n%%\r\nassert(squareOnGrid(4))\r\n\r\n%%\r\nassert(squareOnGrid(5))\r\n\r\n%%\r\nassert(~squareOnGrid(6))\r\n\r\n%%\r\nassert(~squareOnGrid(7))\r\n\r\n%%\r\nassert(squareOnGrid(8))\r\n\r\n%%\r\nassert(squareOnGrid(9))\r\n\r\n%%\r\nassert(squareOnGrid(10))\r\n\r\n%%\r\nassert(~squareOnGrid(11))\r\n\r\n%%\r\nassert(~squareOnGrid(12))\r\n\r\n%%\r\nassert(squareOnGrid(13))\r\n\r\n%%\r\nassert(~squareOnGrid(14))\r\n\r\n%%\r\nassert(~squareOnGrid(15))\r\n\r\n%%\r\nassert(squareOnGrid(16))\r\n\r\n%%\r\nassert(squareOnGrid(17))\r\n\r\n%%\r\nassert(squareOnGrid(18))\r\n\r\n%%\r\nassert(~squareOnGrid(19))\r\n\r\n%%\r\nassert(squareOnGrid(20))\r\n\r\n%%\r\nassert(~squareOnGrid(21))\r\n\r\n%%\r\nassert(~squareOnGrid(42))\r\n\r\n%%\r\nassert(squareOnGrid(49))\r\n\r\n%%\r\nassert(~squareOnGrid(51))\r\n\r\n%%\r\nassert(squareOnGrid(53))\r\n\r\n%%\r\nassert(~squareOnGrid(63))\r\n\r\n%%\r\nassert(squareOnGrid(130))\r\n\r\n%%\r\nassert(squareOnGrid(281))\r\n\r\n%%\r\nassert(~squareOnGrid(283))\r\n\r\n%%\r\nfor k = 566:575\r\n    tf(k-565) = squareOnGrid(k);\r\nend\r\nassert(isequal(sum(tf),1))\r\n\r\n%%\r\nassert(~squareOnGrid(1323))\r\n\r\n%%\r\nassert(~squareOnGrid(2310))\r\n\r\n%%\r\nassert(~squareOnGrid(12768))\r\n\r\n%%\r\nassert(squareOnGrid(12769))\r\n\r\n%%\r\nassert(~squareOnGrid(19200))\r\n\r\n%%\r\nassert(~squareOnGrid(1514205))\r\n\r\n%%\r\nassert(squareOnGrid(2e7))\r\n\r\n%%\r\nassert(squareOnGrid(2^randi(20)))\r\n\r\n%%\r\nassert(~squareOnGrid(7^(2*randi(10)+1)))\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":13,"test_suite_updated_at":"2020-12-25T06:14:57.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-24T15:45:57.000Z","updated_at":"2026-07-20T10:51:05.000Z","published_at":"2020-12-24T16:01:03.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose you have a regular grid of dots, each 1 unit apart. By connecting four dots arranged in a square of side 1, you can draw a square of area 1. Similarly, you can draw squares of area 4, 9, 16, 25, etc. As shown below, by connecting dots with diagonal lines, you can draw squares of area 2, 5, 10, 8, 13, and 17. Which squares can you draw by connecting dots on the grid with straight lines? \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes an area and determines whether the square of that area can be drawn.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.gif\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.gif\",\"contentType\":\"image/gif\",\"content\":\"https://www.mathsisfun.com/activity/images/squares8.gif\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60965,"title":"List the near-square primes, p = n² + 1 (Landau's 4th problem)","description":"Historical context\r\n\r\nAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \"unattackable at the present state of mathematics\" and are now known as Landau's problems (source : Wiki)\r\n\r\nProblem statement\r\n\r\nThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form . Your task here is to list such primes below a given input integer m.\r\n\r\nExamples\r\n\r\nm = 20 =\u003e p = [2, 5, 17] ;\r\nm = 200 =\u003e p = [2, 5, 17, 37, 101, 197] ;\r\nm = 2000 =\u003e p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\r\n\r\nFobidden functions\r\nregexp\r\nstr2num\r\nassignin\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 719.033px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 359.517px; transform-origin: 408px 359.517px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.375px 8px; transform-origin: 375.375px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \"unattackable at the present state of mathematics\" and are now known as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 65.825px 8px; transform-origin: 65.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eLandau's problems \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.725px 8px; transform-origin: 44.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(source : Wiki)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376.275px 8px; transform-origin: 376.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN0AAAAmCAYAAABAkZ7AAAAHVUlEQVR4Xu1cS8huUxg+Z84AIxLC4Mi1XCMGBpQYyb10ilwGkgEh6SQhlGTgEjrJAAMlpRgYKHJJEWJAMWHkFnOep/ZTr2XtvW57r//f3/fuevu////Wevdez1rPe1tr/3v3+OUIOAJdEdjb9W5+M0fAEdjjpPNF4Ah0RsBJ1xlwv50j4KTzNeAIdEbASVcG+Nlo/jDkfMghkF8gd0FeL1PjrbcZASdd/uyTcG9BnoF8DrkScsvQ/Qn8vCdflbfcZgScdPmz/y6avhx4tfvw+yOQvyGH5qvyltuMgJMuf/Y/RtPLIL+ZLofj86/D70cE3+Vr9pZbhYCTrn26/4EK5nanrIR0l+A5H4IwVH60ffgbrYGG9lwzwk/w+bzMET+OdrfHIiAnXSaCI81OwN+/h7wHubRN1eK9RTYtovuddEnMSTpeD0LOhDCVyM3fnXRJeOsa3DZMwsX4+VmdisV7MQS+F3Is5DTIvuGOTro09CKdvNvz6MLiGQ0sDe3U5aRL41vV4mf0ehUyV+WSk3o65CnIXNsQJB0v5qL0diwI8XLSpadc4SXDxOcgxPJDyNGQVOHMSZfGt7gFQaX34ITY4kqxItNBk7wUITaJdBzLTRDrvaewr6kw25zuRCj/AcKto/chH0GmUgonXQsTIn2vwd+4KR5WM1tv46RLI0hv8w7EFjjSveq2dRReHjPc4KjhJ9OKZyFTxnE20rFwwEErf+Hns4YHScW4OcCsoQ0tHTfIbxgs35zP7KSbRlPhHfNSei56m3DvdO75+BMKH4B8CnkDQoPLi+H/1ZBzDB/svatJR5LdCrkCogSc5LoOQqbzpvZS7DvnwOfUJSPBStQZEIaHvJQoE9ADw1g5qWGBRKdSLsR3DDXsxUloDTXXTDpiczyE2F4EoXfYD+F6Ie5cL4wMeHzOLt6S+f12mJtex+84HyQdw0gdhMjN76pJJ0C4GF8bfiFgXHTc6+HnwyBfDGCyyRjzY+Byj6v1KinXc2FcBbFGRCECScNxvQ3R8S67L0MDxHF+A/kgeGgusi8hNFAt15pJx0V5nMFOe5fEjTkQQ0Kbf+VUAC2WCuloDG+GzFVompovSzq2YxGKuaTN70IPKH3NpNOAqZBg7ofYcFKlVH5fUgToTToLyN3DLwSQ+y+85Kn0XCK0DWvGJqnE2IzpWDPpNCZhR4P8JISb8FovXIjCvRQvYfMCdLQatymi2e9C0nEdfA2h9ztpaBh6wNlIp/iVCq+FhFbGgtkTlFzwwnaaQHqyg8OiUFFEG94iIUvFva61k85WR7lODgyi9aJ19B3+fgGkpOorMpd6yJa5C0lHXYyW6N3sOg89INs1e7q/oISx+BhYlnQlnq4FkJa+1pOdCkU2R7OhtMKIlnvZvhanFp2lXkL3WnrLQONTkeNH3Nh6JeVkpTmdfe5a3Gq3DJTT2fvKu8kByQOyjSqcTaQTs6lw7AiMmM42PS1RzQRYUrF/6LmtNVYIUXOfWJ9NJ51IxbEzirBbKjaCiEVLUxjvNtLxWbnm+V4lC3Lav7P5XRPpxGreaMzCyhOu4eCvzT9jRRieMjlywsDMRcCYnjWHl/aNi9hasYUQVo1LQkure+7oY2o+Y+Gl2qfyO1Zyqw88ayGMhZa2yFIaWu5EIUWkYrghKyUgrTXeCY+9ZtLZdRDL6xVBlJzUt4TQvJWusRYjOUU66lUUaCNAeUBWa+np/3dcLOctAxFjrECSIuXUoHuTzpIq5uXk1XfKY6+ZdLbYFouIFA3VkkYRSk1uVku8FOmoVymDjDQ94E8Q1kCiz5oiXViNCquW9s3p3XzSXqDbUDnmyZSbipAcP/che+wJ8Rl3G+k4/t8hOW9QiFQxT2brAiIkSVRS+rcGs1eFPId0mreT8UFhs3hTRTqb/4RvRgtIKub/C1nDMTCRKubJbN7AWJz/B+UVyOWQ8PRJreVM9dtNpNOz8JlTIaElVcyTaR0pReHpFHqD0rczem9N5ZJO3o0HJ+xrQNfXhJe2GmVJxwrgixDe5I5MS5hacD2+lzWOlaytV6cB4VbC/s7GZGnSlSzaMPSfqjjaCCJW6NC4tJXwB3CtPTJnw1iS+M4F5yiXdFy7Wj9cWy9B+AZEcU5nLT89w1cQHnXi+14E7U1Ir7BrDkJaUo2dEZWRoWXn28K9vfdSpCPZ7NE34clx8kjbY5Cwmhhub0zlYqm8Xts0JB29XOx+JXNsI7DcfjW5YAnp+BwhZsXhpa1Gle6r5ALh7f6LgF5iPYg/9zwJk5oHer3aDfmU7trvmeMxPOW5130ZSmpIl6G2vMlUIcW6cf9PV+XYbkoPRghPQ+Y+KLAp+BSPY4p02hdh3OyAF0O7ER1YIGEx6UZITgVzIwa99CDGSGerUb3Ks0uP1fWXIaB/Ic8XOJ1wZdhNth4jnU1Ux4oOMz6Gq3IEtgeBkHRKTlmhtBcrXCRir/2q7ZkBH+nWIZA6kbJ1gPiAHYGlEXDSLY2w63cEAgScdL4kHIHOCPwLWbz3NiLcL+4AAAAASUVORK5CYII=\" width=\"110.5\" height=\"19\" style=\"width: 110.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.775px 8px; transform-origin: 211.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eYour task here is to list such primes below a given input integer\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 78.575px 8px; transform-origin: 78.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 20 =\u0026gt; p = [2, 5, 17] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 125.25px 8px; transform-origin: 125.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 200 =\u0026gt; p = [2, 5, 17, 37, 101, 197] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 230.275px 8px; transform-origin: 230.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 2000 =\u0026gt; p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e I\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = list_the_near_square_primes(m)\r\n  p = m;\r\nend","test_suite":"%%\r\nm = 20;\r\np_correct = [2,5,17];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%%\r\nm = 200;\r\np_correct = [2,5,17,37,101,197];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%%\r\nm = 2000;\r\np_correct = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('list_the_near_square_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":2,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:04:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":63,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-17T04:36:44.000Z","updated_at":"2026-07-20T06:36:47.000Z","published_at":"2025-07-17T05:04:11.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \\\"unattackable at the present state of mathematics\\\" and are now known as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eLandau's problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(source : Wiki)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = n^2 + 1, n \\\\in \\\\mathbb{N}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eYour task here is to list such primes below a given input integer\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 20 =\u0026gt; p = [2, 5, 17] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 200 =\u0026gt; p = [2, 5, 17, 37, 101, 197] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 2000 =\u0026gt; p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":50182,"title":"Compute the Euler totient function","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 123px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.5px; transform-origin: 407px 61.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.7333px 7.91667px; transform-origin: 79.7333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Euler totient function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAAmCAYAAABzhkOMAAACkklEQVRoge1Z25GDMAzcHtIBDdAAFaQCOqCDdEAL1JAS6CEtUENa4D7svQifHzIxDHPjneHHEMmSVg87QEVFRUXF4ejtUwoTgKagvMPQw2y2JBoALwBtYblF0QOYD5LdAlhwUQa0AN44dnM9DAMuhwXA4wQ980l61Ohhon47QdcdwIoL0X8BMP5jfUEwEt2JOkcYphVBB5NHspU0AAa7HuvZE4zxWjRW3+Cs360ud92HIg4fYCj0wjaHRiucm1kRbjML0u2ts9+8rawVn1mgtfpX8aQo3Yj97QIj9nTWHx7Br8C3iKz7QKcyauzdA0yx5DtNO8vRuwENX7Ct0PSom0+zWJffd8iLAPVSzoItdWm8Rt4bOwYq0niFyR2JJ7aUJGbxm2+MX4T8Gdv8pixtLjON1Ljhk3cutRh1n3L+5pvIS/kL/lKWQdEaxICowXxe8beC851rYCt+49Isx3jJON9ARGO0eZxtPGnnUx4qanLTbhvKMZ4p5XM8Eu980HSZX0jaxSjvGigd5rY6VmjNtCUp7+Iu3mvHVh8Tg5AFxd2sjK4cdHqEnUJoqq40zieHLZBBuSF+TqDT1X1e5q5rPCkpo8JWlPLwE+kiJft7aFCSxkyI05+BdLtVFKzabl67Uxc3QMNjUSA7YnRlPfE5UQblbvWmboLozKxTpKQxvSYp2VuBE4yjNLlMCobSQtYTH03le221f2HnVVkHY9RsHzlXj1Y5naDFBH8hA4xxD/uE2MFDjWa4IVOKnCI1eZ1CA8OUM461E3bO9C5SlMwBT4hHokPBe8JQi9uLCcfdsLD7ZFX4GNjiit2KwKRPyT8sCNaiYsipsDlI9elcjIXlbarwEf+ElLzJvcxNbUVFRcXl8AOhCgwX4v7BgwAAAABJRU5ErkJggg==\" alt=\"phi(n)\" style=\"width: 31.5px; height: 19px;\" width=\"31.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.25px 7.91667px; transform-origin: 132.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e gives the number of integers smaller than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 7.91667px; transform-origin: 84.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that are relatively prime to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.3667px 7.91667px; transform-origin: 61.3667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e---that is, that share no common factors with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.95px 7.91667px; transform-origin: 85.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e other than 1. For example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(12) = 4\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 175.283px 7.91667px; transform-origin: 175.283px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because 1, 5, 7, and 11 share no common factors other than 1. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.0083px 7.91667px; transform-origin: 77.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a prime number, then \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(p) = p-1\" style=\"width: 85px; height: 19px;\" width=\"85\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.533px 7.91667px; transform-origin: 108.533px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because all numbers smaller than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.7917px 7.91667px; transform-origin: 70.7917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are relatively prime to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 166.333px 7.91667px; transform-origin: 166.333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the Euler totient function. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNote: I now realize that \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 656\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.517px 7.91667px; transform-origin: 174.517px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e also deals with the totient function. Apologies to Freddy.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function phi = totient(n)\r\n   phi = f(n)\r\nend","test_suite":"%%\r\nn = 1;\r\nphi_correct = 1;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 2;\r\nphi_correct = 1;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 3;\r\nphi_correct = 2;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 4;\r\nphi_correct = 2;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 6;\r\nphi_correct = 2;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 12;\r\nphi_correct = 4;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 17;\r\nphi_correct = 16;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 44;\r\nphi_correct = 20;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 100;\r\nphi_correct = 40;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 256;\r\nphi_correct = 128;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 729;\r\nphi_correct = 486;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 2310;\r\nphi_correct = 480;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 4999;\r\nphi_correct = 4998;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 12345;\r\nphi_correct = 6576;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 654321;\r\nphi_correct = 436212;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 7566179;\r\nphi_correct = 7451136;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 901800900;\r\nphi_correct = 172972800;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 7956061979;\r\nphi_correct = 7944111936;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nassert(isequal(totient(totient(6503)),1792))\r\n\r\n%%\r\np = primes(2000);\r\nk = randi(length(p));\r\nassert(isequal(totient(p(k)),p(k)-1))\r\n\r\n%%\r\nP = primes(2000);\r\np = P(randi(length(P)));\r\nq = P(randi(length(P)));\r\nassert(isequal(totient(p*q),(p-1)*(q-1)))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-02-06T18:48:05.000Z","updated_at":"2026-07-14T12:34:15.000Z","published_at":"2021-02-06T19:13:15.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Euler totient function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e gives the number of integers smaller than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that are relatively prime to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e---that is, that share no common factors with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e other than 1. For example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(12) = 4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(12) = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because 1, 5, 7, and 11 share no common factors other than 1. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a prime number, then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(p) = p-1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(p) = p-1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because all numbers smaller than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are relatively prime to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the Euler totient function. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNote: I now realize that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 656\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e also deals with the totient function. Apologies to Freddy.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"problems":[{"id":61269,"title":"Precise Almost Pythagorean Triples ","description":"This  is essentially the same as:  Problem 52834. Easy Sequences 32: Almost Pythagorean Triples; it even presents the same set of test problems.  The difference is that the \"correct\" solutions for larger cases of 52834 were the result of roundoff errors due to values being larger than flintmax('double').  This problem requires care to avoid such roundoff.\r\nRepeating the original problem description:\t\t\t\t\r\nAn Almost Pythagorean Triple (abbreviated as \"APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is 1 less than the sum of square of the smaller elements (shorter sides). This means that if c is the hypotenuse and a and b are the shorter sides, , satisfies the following equation: \r\n        \r\n        where:  \r\nThe smallest  is the triple , with  and perimeter (the sum of the 3 elements)  of . Some researchers consider  as the smallest , but here, we will only look at 's where the hypotenuse is \"strictly\" greater than the other shorter sides. Other examples of 's are , and . \r\nUnfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible 's, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with 's with a known ratio between the hypotenuse and the shortest side: . \r\nGiven the value of r, find the perimeter of the  with the r-th smallest perimeter. For example for , that is , the smallest perimeter is  for  , while the second (r-th) smallest perimeter is , for the  with dimensions . For , the third smallest perimeter is  for  . \r\nThe output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.\r\nFinally, as with the original, the use of java, BigInteger, persistent, and global are not allowed.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 669.833px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 334px 334.917px; transform-origin: 334px 334.917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 42px; text-align: left; transform-origin: 310px 42px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 102.675px 7.91667px; transform-origin: 102.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis  is essentially the same as:  \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/52834\"\u003e\u003cspan style=\"border-block-end-color: rgb(0, 91, 130); border-block-start-color: rgb(0, 91, 130); border-bottom-color: rgb(0, 91, 130); border-inline-end-color: rgb(0, 91, 130); border-inline-start-color: rgb(0, 91, 130); border-left-color: rgb(0, 91, 130); border-right-color: rgb(0, 91, 130); border-top-color: rgb(0, 91, 130); caret-color: rgb(0, 91, 130); color: rgb(0, 91, 130); column-rule-color: rgb(0, 91, 130); outline-color: rgb(0, 91, 130); text-decoration-color: rgb(0, 91, 130); text-emphasis-color: rgb(0, 91, 130); \"\u003e\u003cspan style=\"\"\u003eProblem 52834. Easy Sequences 32: Almost Pythagorean Triples\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e; it even presents the same set of test problems.  The difference is that the \"correct\" solutions for larger cases of 52834 were the result of roundoff errors due to values being larger than flintmax('double').  This problem requires care to avoid such roundoff.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 10.5px; text-align: left; transform-origin: 310px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 201.933px 7.91667px; transform-origin: 201.933px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eRepeating the original problem description:\t\t\t\t\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 86.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 43.4583px; text-align: left; transform-origin: 310px 43.4583px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 303.05px 7.91667px; transform-origin: 303.05px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn Almost Pythagorean Triple (abbreviated as \"APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 7.91667px; transform-origin: 3.89167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.45px 7.91667px; transform-origin: 12.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eless\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.8917px 7.91667px; transform-origin: 94.8917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e than the sum of square of the smaller elements (shorter sides). This means that if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 7.91667px; transform-origin: 3.5px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ec\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.35px 7.91667px; transform-origin: 72.35px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the hypotenuse and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 7.91667px; transform-origin: 3.89167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ea\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 7.91667px; transform-origin: 3.89167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eb\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.3917px 7.91667px; transform-origin: 49.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are the shorter sides, \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 102.692px 7.91667px; transform-origin: 102.692px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, satisfies the following equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 24.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 12.4583px; text-align: left; transform-origin: 310px 12.4583px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5333px 7.91667px; transform-origin: 15.5333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"99\" height=\"19\" style=\"vertical-align: baseline;width: 99px;height: 19px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 23.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 11.9583px; text-align: left; transform-origin: 310px 11.9583px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.425px 7.91667px; transform-origin: 40.425px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        where:  \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"62\" height=\"18\" style=\"vertical-align: baseline;width: 62px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 96.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 48.3333px; text-align: left; transform-origin: 310px 48.3333px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.6167px 7.91667px; transform-origin: 41.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe smallest \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.725px 7.91667px; transform-origin: 37.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the triple \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"48\" height=\"18\" style=\"vertical-align: baseline;width: 48px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGEAAAAlCAYAAABSz4fZAAACsklEQVRoge2a0ZGDIBRFTw92YAM2kApSQTqwAztIC9awJdhDWrCGtJD9wDcSFQTDI+4OZ4afDAJ5Fy48FAqFQuH0dI5y+eag/gAX3LGL5gU8gGFRbilG+o+5sY7ZAxPPaF6UWZ+KC5lEqKb6W+UK3I8MwkPj6a8H6g/b9/2frVJ52somQjc94yopV1UFPD19DQn6uHnaX5YnftGziJAjKDY5BBcfDyl7/y+LCB0w4l+SqRDBW8U+JGh7h5B7YL0sIjxJ7/kuWvaX/6f0hIk8Bo5FXQTxzhazWWozYqxibzP8hJBjeEO41aqLMLLepHp0BNnaLB/T7zms0EasKCQJUxVh7wTRkzY4S8GX4ufMa2QsIbaovhLqqV4L/LB9ckgphJ13bIlyTdiXC7GiR2D9bHmCULMW49C9SSA33o/HT/StKcaK4AsiCLZVaQem4v1cryk6xFkRfFEEmGdMjnuomnlFpE4ObWKtCL4sQs0sgmZyJfTMK0+LWCuCL4sAs03kEOGKvgixVgQnEGEgLLVPgfzZUbn9GCuyn4sm9UrQvGYQZCX0Su2L3cVu/KfYE2JnzlEkSFq5gmz8sRNKVYRmp44EZatOhdknOsKuOOQliysAIrjrZCTvfo9eccgqO2J1aiK0zCefkXUgJU9wLV0RSMqeEHZC1vEeSMkTHo527FPaC5NMxiLjPXJbrCaCNLy8K+ows3HEbwvLrHrPxwfe649TX3fmS0PXDG9YjzXWUmQSHLFoVTuqMUGwvyyQJb9HgxFiYA7iHjfrmYFZ9JCAyuQ4clBopmePrCA4wekohJyf0lzQO8L6+jy1CA35Xo2Cmc05EkebU4vQ4N5MNejJ9xrW5iMRWtbf16RKuipMQHIkcWDsLoflyfsVu8hJMhrXGyzta+K/ju9znEKhUCgUCmfnFzTbYsvcMy9LAAAAAElFTkSuQmCC\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.275px 7.91667px; transform-origin: 18.275px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, with \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"100\" height=\"19\" style=\"vertical-align: baseline;width: 100px;height: 19px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMkAAAAmCAYAAABu1L9SAAAD6ElEQVR4nO2c7ZGjMAyG3x7SAQ3QABWkAjpIB3SQFqghJdBDWqCGtLD3w7yDz3zYAdlgomdGczO7XGJblixLYgFFURRFURRFURRFUX6KO4AOwN8gHYDy0BGloQTwAvCBmXcPsxZX5Fd1LEIN4D38e4fZNFzI6sBxxaaEMYoHzDyfGOfdHDiuGPyqjsXoARTOz7iIr/TDScYL01PjATPvT/rhROVXdTyhhPEKIcJjlh7U5Q6zgO+4QxanwPKcHzCelHQAbs7/v2H0sO7vzsaavluMRnFFHW/WTY9RwT6ZWzSbCnl6GTvmduWDsMXl3eTMRnLDeI+aky7gM3LTcQFj/H/YeJeiVwgVXxxaD889tgzmIKj0JQm5ZxTIY+M02KdfIB8d3zAaB2WTkXSDuHGn+2X0kj5amGP4zN7UpcP+zV3DeOgzZ314iuzd3Dno+A7jEO4wY91sJAXCwgN6Dl+oRW+aU9ajhBnz3vRtD9nM1i7Pt8ADxkjWHKKPHHVsn56bjCRkssxm+J7tcP4j2IXHcYXtm6cZPkfSs8Ywkh7Gq1bYPtYcdbzLSEJgqOW7vD7hP2nOBr2iLTwRQg2GhTbp0ENaqYwGbGH9I3TsOeoYSGAkXNzW84y0J02Be6lzxecxS/jvc1uRVupaBvODsIRMjjoGEhgJQ62lmL3Gct1gzbDcZ0PrNGuyZbOyZtDg/wueL+XNqvvcd0psphhKrWD0+MS80cTU8ZFENRJfqMX0cTsMxJZvepl8KdhQkYiVK0yNxfWyBcyadJjOu4NMSBLV8w0wI7dWD5LS8ZFENZK1UKvEelHqm6Iaw5a9IqkwOxSzC2w3zJ84u3PxDimMBJjOx87QSer4SKIaiS/UujrcPKEV92/wtQJRqbXnOQl4MroOQQqJcHrP5o5mJKFZrStjZ4SkPdBaG0yoSDZT8uSM0aApEU7vKfZGM5KQrNbVsZsWpY3kifXQkd/7XnlGsgWGd48YRiIRTu8p1EYzEioqRah1ZHbLBxc3xmeHfG+qNhd6+5DWo9yIYiQssqUKtc6U3bL5pmdNmtRGYmexrkYUI+ELRKkW7IzZLSC8Zy0GqY2Ed5IrJmmiGAmzOrXvwczxZU16LBcMYyOpVIazS/Ng5BAjs3UGxI3EDrWO2BypsN/VfmE61xZmDY7yrJJKtWsdDf4PoVkneQt91xkRNxKGWlf1KsR9+egDE1axknz0XwSRVKqbbmYT5xNm3rn2ZIXA10HmiqWbYWryirGpC9/ptlOqDc7xroR0DF3DzI9zZZvJVaOFAstp9hb5tfsrM7A/6qqbWFEURVEURVEURVEURVEURVEURTkZ/wCi5zysuW8z8wAAAABJRU5ErkJggg==\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 101.508px 7.91667px; transform-origin: 101.508px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and perimeter (the sum of the 3 elements)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 7.91667px; transform-origin: 7.775px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eof \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"18\" height=\"18\" style=\"vertical-align: baseline;width: 18px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA6UlEQVRYhe2WQQ3DMAxFP4cwCIESGIIiKIMxKINSKIZBKIdRKIZR6A6ztWiHxkm+qkjzl6xenOQpL5UMeDyevhIBhEzPAOBmrKEFZAVwGDbZpc9SSylISEC0zoDGApgDn1syZwQwy/dpBNqk4klPkH32EpjfzAagKIfk3tiESl01QBYFD1ToqgGyRHW9kL/JS4BU19oCwwRSXWMPQDRdLCCaLhYQTRcDiKqLAUTVxQDaQNTVChRB1tUKdAdZVyuQTgpTD0CprrORpCg6VijQXLBWdW0skAXfoSutVQ7LRdfT/i6Px+P5u7wB3c96XQFb71kAAAAASUVORK5CYII=\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91.4083px 7.91667px; transform-origin: 91.4083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Some researchers consider \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"48\" height=\"18\" style=\"vertical-align: baseline;width: 48px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.5583px 7.91667px; transform-origin: 50.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as the smallest \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.7917px 7.91667px; transform-origin: 70.7917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, but here, we will only look at \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 243.942px 7.91667px; transform-origin: 243.942px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e's where the hypotenuse is \"strictly\" greater than the other shorter sides. Other examples of \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.8333px 7.91667px; transform-origin: 18.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e's are \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"56\" height=\"18\" style=\"vertical-align: baseline;width: 56px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 7.91667px; transform-origin: 17.5px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"71\" height=\"18\" style=\"vertical-align: baseline;width: 71px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 71.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 35.875px; text-align: left; transform-origin: 310px 35.875px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 269.142px 7.91667px; transform-origin: 269.142px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eUnfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.9417px 7.91667px; transform-origin: 21.9417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e's, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.775px 7.91667px; transform-origin: 6.775px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e's with a known ratio between the hypotenuse and the shortest side: \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"50\" height=\"18\" style=\"vertical-align: baseline;width: 50px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 95.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 47.8333px; text-align: left; transform-origin: 310px 47.8333px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.2083px 7.91667px; transform-origin: 62.2083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven the value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2.33333px 7.91667px; transform-origin: 2.33333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003er\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.925px 7.91667px; transform-origin: 85.925px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, find the perimeter of the \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3167px 7.91667px; transform-origin: 30.3167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e with the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.6167px 7.91667px; transform-origin: 41.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; text-decoration-line: underline; \"\u003er-th smallest\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.3417px 7.91667px; transform-origin: 37.3417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e perimeter. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.4417px 7.91667px; transform-origin: 12.4417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example for \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"35\" height=\"18\" style=\"vertical-align: baseline;width: 35px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.4917px 7.91667px; transform-origin: 24.4917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, that is \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"43\" height=\"18\" style=\"vertical-align: baseline;width: 43px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.675px 7.91667px; transform-origin: 81.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the smallest perimeter is \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"18\" height=\"18\" style=\"vertical-align: baseline;width: 18px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAABLUlEQVRYhe2WUbHDIBBFjwccYKAGoiAK4qAOcBAL0RAJeKiFaoiFvg+WmX2dPEIC5fWDO7Nfu1lOLiwJdHV1fZcsYE4+M0jcaoMswCuzsQFm4Ak4iafEvQTEKJAYR0AWeMji726u0sPv5A41Et5slAVygWKt28kZYEvks+UygaaMOu34aZfOAsUt2RI1GvqyS7lAcTtSQIPqtX4aKNakgGxmXVWgo/PRDMiruvEbgO4cn4+mW2YIF2LKJT32j08DIXkN5eX5meDIqnJLCyAITk2yoBcIR9iumbxzVhUoBRrvqr1vXXMg3Wcq6FMF6MbxBDYD0tN36dejJpCeurUGjOX3GJ/5QjvCId4o/FOMIDPB4vdY/ljAEEY53jle6opduSorAMN/QnR1dXWV6Ae9sZqCc2SAGwAAAABJRU5ErkJggg==\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.05px 7.91667px; transform-origin: 12.05px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"48\" height=\"18\" style=\"vertical-align: baseline;width: 48px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.9583px 7.91667px; transform-origin: 57.9583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, while the second (r-th) smallest perimeter is \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"18\" height=\"18\" style=\"vertical-align: baseline;width: 18px;height: 18px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA6klEQVRYhe2WUQ2DMBCGfw91UAMYmIIqwMEc4AAL1TAJ9TALaJgF9sBd0i2DXQ9u4+G+5MIDBT74QgBwHOc/dAAuwumE54wAglZoAjALZxSIZForlX8hNcjMWJ7SJ0IlwqMSKjRxY02gC0wr+xOAgbb3PUKRLvKtdQ9ZLpDYLqG1BDU3bOc6TEgC53pA9taYC3GuLFxvLsS50hmEWnOZC7XmMhdqzWUqpMllKqTJZSpU0J7LTChCl8tM6ApdLjMh/mL3ZxCqc239kvxMiHMVxbH8O8NCwxFCI8m0vF2xOu59MpabdBzHcSQ8AUZlel3qgGXQAAAAAElFTkSuQmCC\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.6583px 7.91667px; transform-origin: 25.6583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, for the \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.6833px 7.91667px; transform-origin: 53.6833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with dimensions \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"71\" height=\"18\" style=\"vertical-align: baseline;width: 71px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.325px 7.91667px; transform-origin: 16.325px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"35\" height=\"18\" style=\"vertical-align: baseline;width: 35px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.1083px 7.91667px; transform-origin: 31.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the third smallest perimeter is \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"33\" height=\"18\" style=\"vertical-align: baseline;width: 33px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.05px 7.91667px; transform-origin: 12.05px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"32\" height=\"18\" style=\"vertical-align: baseline;width: 32px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"101\" height=\"18\" style=\"vertical-align: baseline;width: 101px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 21px; text-align: left; transform-origin: 310px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 305.717px 7.91667px; transform-origin: 305.717px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 310px 21px; text-align: left; transform-origin: 310px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 282.908px 7.91667px; transform-origin: 282.908px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFinally, as with the original, the use of java, BigInteger, persistent, and global are not allowed.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function function perimeter = rthPerAPTdbl(r)\r\n  perimeter = r^2;\r\nend","test_suite":"%% Test Case 1\r\nr = 2;\r\np_correct = 71;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 2\r\nr = 3;\r\np_correct = 1393;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 3\r\nr = 5;\r\np_correct = 1046629;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 4\r\nr = 10;\r\np_correct = 737287485879;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 5\r\nr = 100;\r\np_correct = 16183149010201;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 6\r\nrs = 101:150;\r\nps = arrayfun(@(r) rthPerAPTdbl(r),rs);\r\nps = mod([sum(ps) ps(5:5:end) floor(std(double(ps)))],1e6);\r\nps_correct = [12636 824229 203679 227761 926641 15749 664839 210241 515881 139269 477199 789840];\r\nassert(isequal(ps,ps_correct))\r\n%% Test Case 7\r\nr = 1000;\r\np_correct = 499499001002001;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 8\r\nr = 10000;\r\np_correct = 100020001;\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 9\r\nr = 123456;\r\np_correct = uint64(76696064606196865);\r\nassert(isequal(rthPerAPTdbl(r),p_correct))\r\n%% Test Case 10: Forbid java and BigInteger\r\nfiletext = fileread('rthPerAPTdbl.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'BigInteger') || contains(filetext, 'java'); \r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":5,"created_by":2404920,"edited_by":2404920,"edited_at":"2026-03-04T15:24:47.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2026-03-04T14:14:29.000Z","updated_at":"2026-07-19T13:59:15.000Z","published_at":"2026-03-04T15:24:47.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis  is essentially the same as:  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/52834\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 52834. Easy Sequences 32: Almost Pythagorean Triples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e; it even presents the same set of test problems.  The difference is that the \\\"correct\\\" solutions for larger cases of 52834 were the result of roundoff errors due to values being larger than flintmax('double').  This problem requires care to avoid such roundoff.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eRepeating the original problem description:\\t\\t\\t\\t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn Almost Pythagorean Triple (abbreviated as \\\"APT'), is a set of 3 integers in which square of the largest element, which we will call as its 'hypotenuse', is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eless\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e than the sum of square of the smaller elements (shorter sides). This means that if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the hypotenuse and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are the shorter sides, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, satisfies the following equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"19\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"99\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        where:  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"62\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe smallest \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the triple \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"48\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"19\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"100\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and perimeter (the sum of the 3 elements)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eof \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId6\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Some researchers consider \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"48\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId7\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the smallest \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, but here, we will only look at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e's where the hypotenuse is \\\"strictly\\\" greater than the other shorter sides. Other examples of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e's are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"56\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId8\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"71\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId9\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUnfortunately, unlike Pythagorean Triples, a 'closed formula' for generating all possible \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e's, has not yet been discovered, at the time of this writing. For this exercise, we will be dealing with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e's with a known ratio between the hypotenuse and the shortest side: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"50\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId10\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven the value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, find the perimeter of the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e with the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er-th smallest\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e perimeter. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"35\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId11\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, that is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"43\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId12\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the smallest perimeter is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId13\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"48\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId14\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, while the second (r-th) smallest perimeter is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId15\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, for the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with dimensions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"71\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId16\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"35\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId17\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the third smallest perimeter is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"33\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId18\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"32\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"101\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId19\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe output can be very large, so please present only the last 12 digits if the number of digits of the perimeter exceeds 12.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFinally, as with the original, the use of java, BigInteger, persistent, and global are not 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sum","description":"Given natural number calculate its _population count_.","description_html":"\u003cp\u003eGiven natural number calculate its \u003ci\u003epopulation count\u003c/i\u003e.\u003c/p\u003e","function_template":"function population = popcount(n)\r\n  binary = dec2bin(n);\r\n  population = ???\r\nend","test_suite":"%%\r\nN = 1; \r\nnum_ones = 1;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 2; \r\nnum_ones = 1;\r\nassert(isequal(popcount(N),num_ones))\r\n%%\r\nN = 3; \r\nnum_ones = 2;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nfor num_ones = 3:10\r\n  N = 2^num_ones-1;\r\n  assert(isequal(popcount(N),num_ones))\r\nend\r\n\r\n%%\r\n\r\nfor num_ones = 1:50\r\n  N = sum(2.^(find(randperm(50)\u003c=num_ones)-1));\r\n  assert(isequal(popcount(N),num_ones))\r\nend\r\n\r\n%%\r\nN = 23; \r\nnum_ones = 4;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 123; \r\nnum_ones = 6;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 123456; \r\nnum_ones = 6;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 1234567; \r\nnum_ones = 11;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n%%\r\nN = 1125899906842501;\r\nnum_ones = 45;\r\nassert(isequal(popcount(N),num_ones))\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":76,"test_suite_updated_at":"2015-01-19T23:10:17.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2014-12-08T11:59:42.000Z","updated_at":"2026-03-09T20:12:46.000Z","published_at":"2015-01-19T11:07:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven natural number calculate its\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epopulation count\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46000,"title":"Compute the harmonic numbers","description":"The nth \u003chttps://mathworld.wolfram.com/HarmonicNumber.html harmonic number\u003e is defined as the sum of the reciprocals of the integers from 1 to n: \r\n\r\n\u003c\u003chttps://mathworld.wolfram.com/images/equations/HarmonicNumber/NumberedEquation1.gif\u003e\u003e\r\n\r\nThe first four harmonic numbers are 1, 3/2, 11/6, and 25/12. The harmonic numbers appear in several interesting identities, including this one for \u003chttps://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant Apery's constant\u003e:\r\n\r\n\u003c\u003chttps://mathworld.wolfram.com/images/equations/RiemannZetaFunction/NumberedEquation34.gif\u003e\u003e\r\n\r\nCompute the nth harmonic number.\r\n\r\n\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 210px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 105px; transform-origin: 407px 105px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 7.8px; transform-origin: 14px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 7.8px; transform-origin: 5.83333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/HarmonicNumber.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eharmonic number\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 201.083px 7.8px; transform-origin: 201.083px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined as the sum of the reciprocals of the integers from 1 to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Definition of the harmonic numbers: sum(1/k) from k = 1 to k = n\" style=\"width: 69px; height: 45px;\" width=\"69\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 381.85px 7.8px; transform-origin: 381.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe first four harmonic numbers are 1, 3/2, 11/6, and 25/12. The harmonic numbers appear in several interesting identities, including this one for\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eApery's constant\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Relation between zeta(3) and the harmonic numbers\" style=\"width: 98.5px; height: 45px;\" width=\"98.5\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 42.0167px 7.8px; transform-origin: 42.0167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCompute the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.8px 7.8px; transform-origin: 63.8px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth harmonic number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function H = HarmonicNumber(n)\r\n  H = f(n);\r\nend","test_suite":"%%\r\nn = 1;\r\nH_correct = 1;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 2;\r\nH_correct = 3/2;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 3;\r\nH_correct = 11/6;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 4;\r\nH_correct = 25/12;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 5;\r\nH_correct = 137/60;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 10;\r\nH_correct = 7381/2520;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 15;\r\nH_correct = 1195757/360360;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 20;\r\nH_correct = 55835135/15519504;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nn = 29;\r\nH_correct = 9227046511387/2329089562800;\r\nassert(abs(HarmonicNumber(n)-H_correct)/H_correct \u003c 1e-12)\r\n\r\n%%\r\nfiletext = fileread('HarmonicNumber.m');\r\nx = ~isempty(strfind(filetext, 'A002805')) || ~isempty(strfind(filetext, 'A001008')) ||...\r\n    ~isempty(strfind(filetext, '360360'));\r\nassert(~x, 'Illegal approach')","published":true,"deleted":false,"likes_count":4,"comments_count":1,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":96,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-29T03:02:15.000Z","updated_at":"2026-07-09T05:57:38.000Z","published_at":"2020-06-29T03:50:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/HarmonicNumber.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eharmonic number\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is defined as the sum of the reciprocals of the integers from 1 to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Definition of the harmonic numbers: sum(1/k) from k = 1 to k = n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eH_n = \\\\sum_{k = 1}^n\\\\frac{1}{k}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first four harmonic numbers are 1, 3/2, 11/6, and 25/12. The harmonic numbers appear in several interesting identities, including this one for\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eApery's constant\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Relation between zeta(3) and the harmonic numbers\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\zeta(3) = \\\\frac{1}{2}\\\\sum_{k = 1}^\\\\infty\\\\frac{H_k}{k^2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth harmonic number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":43082,"title":"Reverse digits of a number","description":"Reverse digits of a number.\r\n\r\nFor example, return *321* for the input *123* .\r\n\r\nThe left-hand zeros can be discarded. For example, input *10* will return *1* .","description_html":"\u003cp\u003eReverse digits of a number.\u003c/p\u003e\u003cp\u003eFor example, return \u003cb\u003e321\u003c/b\u003e for the input \u003cb\u003e123\u003c/b\u003e .\u003c/p\u003e\u003cp\u003eThe left-hand zeros can be discarded. For example, input \u003cb\u003e10\u003c/b\u003e will return \u003cb\u003e1\u003c/b\u003e .\u003c/p\u003e","function_template":"function y = reverseDigits(x)\r\n    y = x;\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = 1;\r\nassert(isequal(reverseDigits(x),y_correct))\r\n\r\n%%\r\nx = 10;\r\ny_correct = 1;\r\nassert(isequal(reverseDigits(x),y_correct))\r\n\r\n%%\r\nx = 675;\r\ny_correct = 576;\r\nassert(isequal(reverseDigits(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":25354,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":77,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-10-05T22:52:12.000Z","updated_at":"2026-05-28T16:01:59.000Z","published_at":"2016-10-05T22:52:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReverse digits of a number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e321\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for the input\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e123\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e .\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe left-hand zeros can be discarded. For example, input\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e .\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2369,"title":"Tribute to Ramanujan","description":"The nth taxicab number, denoted as T(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways (source: wikipedia).\r\n\r\nExample: \r\n\r\n           T(1) = 2 = 1^3 + 1^3\r\n           T(2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3\r\n\r\nReturn the value of n if the given number is a taxicab number. Return 0 if not. If the input is 1729, output would be 2, since 1729 is smallest number that can be expressed as a sum of two cubes in two (n=2) distinct ways.\r\n\r\nAvoid look up table solution. Test suit might expand.\r\n\r\n","description_html":"\u003cp\u003eThe nth taxicab number, denoted as T(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways (source: wikipedia).\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e           T(1) = 2 = 1^3 + 1^3\r\n           T(2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3\u003c/pre\u003e\u003cp\u003eReturn the value of n if the given number is a taxicab number. Return 0 if not. If the input is 1729, output would be 2, since 1729 is smallest number that can be expressed as a sum of two cubes in two (n=2) distinct ways.\u003c/p\u003e\u003cp\u003eAvoid look up table solution. Test suit might expand.\u003c/p\u003e","function_template":"function y = mytaxicab(x)\r\n\r\n\r\n\r\nend","test_suite":"%%\r\nx = 2;\r\ny_correct = 1;\r\nassert(isequal(mytaxicab(x),y_correct))\r\n\r\n%%\r\nx = 1729;\r\ny_correct = 2;\r\nassert(isequal(mytaxicab(x),y_correct))\r\n\r\n%%\r\nx = 5251;\r\ny_correct = 0;\r\nassert(isequal(mytaxicab(x),y_correct))\r\n\r\n%%\r\nx = 10000001;\r\ny_correct = 0;\r\nassert(isequal(mytaxicab(x),y_correct))\r\n\r\n\r\n%%\r\n\r\nx = 87539319;\r\ny_correct = 3;\r\nassert(isequal(mytaxicab(x),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":6,"created_by":17203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":33,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2014-06-18T06:26:28.000Z","updated_at":"2026-05-28T02:36:30.000Z","published_at":"2014-06-18T06:27:24.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe nth taxicab number, denoted as T(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways (source: wikipedia).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[           T(1) = 2 = 1^3 + 1^3\\n           T(2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn the value of n if the given number is a taxicab number. Return 0 if not. If the input is 1729, output would be 2, since 1729 is smallest number that can be expressed as a sum of two cubes in two (n=2) distinct ways.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAvoid look up table solution. Test suit might expand.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46305,"title":"find the value","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"display: block; min-width: 0px; padding-top: 0px; vertical-align: baseline; perspective-origin: 407px 10.5px; transform-origin: 407px 10.5px; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; perspective-origin: 384px 10.5px; transform-origin: 384px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003epi*(2+x)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = your_fcn_name(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 5;\r\ny_correct = 7*pi;\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":424112,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":116,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-08-14T07:42:35.000Z","updated_at":"2026-05-30T15:28:23.000Z","published_at":"2020-08-14T07:42:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003epi*(2+x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":43130,"title":"Nth root","description":"Nth root of a number x","description_html":"\u003cp\u003eNth root of a number x\u003c/p\u003e","function_template":"function y = nroot(n,x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 1;\r\nn=1;\r\ny_correct = 1;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n%%\r\nx = 1;\r\nn=2;\r\ny_correct = 1;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n\r\n%%\r\nx = 1;\r\nn=2;\r\ny_correct = 1;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n\r\n%%\r\nx = 10;\r\nn=2;\r\ny_correct = 3.1623;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n%%\r\nx = 10;\r\nn=10;\r\ny_correct = 1.2589;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n\r\n%%\r\nx = 100;\r\nn=100;\r\ny_correct = 1.0471;\r\nassert((nroot(n,x)-y_correct)\u003c0.001)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":91311,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":138,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-10-06T17:19:12.000Z","updated_at":"2026-02-17T09:14:29.000Z","published_at":"2016-10-06T17:19:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNth root of a number x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":42865,"title":"Find the sum of n squares","description":"What is the sum of the squares of the first n integers?","description_html":"\u003cp\u003eWhat is the sum of the squares of the first n integers?\u003c/p\u003e","function_template":"function y = your_fcn_name(x)\r\n  y = x^2;\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = 1;\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n\r\nx = 15;\r\ny_correct = 1240;\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n\r\nx = 50;\r\ny_correct = 42925;\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":69320,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":210,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2016-06-02T15:33:45.000Z","updated_at":"2026-07-07T03:36:09.000Z","published_at":"2016-06-02T15:33:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the sum of the squares of the first n integers?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":54370,"title":"Determine whether the given number is palindrome or not.","description":"A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed. \r\nFor more: https://en.wikipedia.org/wiki/Palindromic_number#:~:text=A%20palindromic%20number%20(also%20known,when%20its%20digits%20are%20reversed.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003epalindromic number\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (also known as a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enumeral palindrome\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enumeric palindrome\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e) is a number (such as 16461) that remains the same when its digits are reversed. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor more: https://en.wikipedia.org/wiki/Palindromic_number#:~:text=A%20palindromic%20number%20(also%20known,when%20its%20digits%20are%20reversed.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = palindrome(x)\r\ny=x\r\nend","test_suite":"%%\r\nx = 16465;\r\ny_correct = false;\r\nassert(isequal(palindrome(x),y_correct))\r\n%%\r\nx = 16461;\r\ny_correct = true;\r\nassert(isequal(palindrome(x),y_correct))\r\n%%%\r\nx = 12321;\r\ny_correct = true;\r\nassert(isequal(palindrome(x),y_correct))\r\n%%%%%\r\nx = 54353453;\r\ny_correct = false;\r\nassert(isequal(palindrome(x),y_correct))\r\n%%%%%\r\nx = 88888888888888;\r\ny_correct = true;\r\nassert(isequal(palindrome(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":1780615,"edited_by":1780615,"edited_at":"2022-04-27T16:41:00.000Z","deleted_by":null,"deleted_at":null,"solvers_count":34,"test_suite_updated_at":"2022-04-27T16:41:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-04-27T16:18:37.000Z","updated_at":"2026-06-01T04:34:54.000Z","published_at":"2022-04-27T16:41:00.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epalindromic number\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (also known as a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumeral palindrome\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e or a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumeric palindrome\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e) is a number (such as 16461) that remains the same when its digits are reversed. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor more: https://en.wikipedia.org/wiki/Palindromic_number#:~:text=A%20palindromic%20number%20(also%20known,when%20its%20digits%20are%20reversed.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46639,"title":"Transform a Decimal number into a Septimal number ","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 211.4px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 105.7px; transform-origin: 407px 105.7px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 62.4px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.2px; text-align: left; transform-origin: 384px 31.2px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSome scientists believes that a septimal system (7 base number system) was in use approximately 2500 years ago by the Magyar (Hungarian) tribes. Though some are disagreed. However, How will it be if you try to convert some numbers to their number sytem (Septimal Number System) from ours (Decimal Number System). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eLet me it crystal clear by incluing an example.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eLet,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eDecimal number, x = 93 (as Input)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSeptical Number, y = 162 (as Output)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e which is equivalent to the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eDecimal number, x = 93 (Input)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAs you have to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003econvert a decimal number into a septimal number\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e ,you must show the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eoutput as y\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (a septimal number).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y=DecimaltoSeptimal(x)\r\ny=[];\r\nend\r\n","test_suite":"%%\r\nx =7;\r\ny= 10;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =93;\r\ny= 162;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =22;\r\ny= 31;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =20145;\r\ny= 112506;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =2014;\r\ny= 5605;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n%%\r\nx =107033;\r\ny= 624023;\r\nassert(isequal(DecimaltoSeptimal(x),y))\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":430818,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-10-01T19:42:08.000Z","updated_at":"2026-05-30T17:40:40.000Z","published_at":"2020-10-01T19:49:16.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSome scientists believes that a septimal system (7 base number system) was in use approximately 2500 years ago by the Magyar (Hungarian) tribes. Though some are disagreed. However, How will it be if you try to convert some numbers to their number sytem (Septimal Number System) from ours (Decimal Number System). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet me it crystal clear by incluing an example.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDecimal number, x = 93 (as Input)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSeptical Number, y = 162 (as Output)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e which is equivalent to the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDecimal number, x = 93 (Input)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs you have to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003econvert a decimal number into a septimal number\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ,you must show the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eoutput as y\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (a septimal number).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":2370,"title":"Fermat's last theorem - again","description":"For a given integer n, express it as a sum of two squares if possible. Return empty matrix otherwise. Solution may not be unique and uniqueness will not be checked, so return one of the solutions.\r\nExample: \r\n\r\n  n=25\r\n  Output=[3 4]\r\n\r\n\r\n  n=41\r\n  Output=[5 4]\r\n \r\n  n=7\r\n  Output=[]","description_html":"\u003cp\u003eFor a given integer n, express it as a sum of two squares if possible. Return empty matrix otherwise. Solution may not be unique and uniqueness will not be checked, so return one of the solutions.\r\nExample:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003en=25\r\nOutput=[3 4]\r\n\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003en=41\r\nOutput=[5 4]\r\n\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003en=7\r\nOutput=[]\r\n\u003c/pre\u003e","function_template":"function y = sumOfSqr(x)\r\n\r\n\r\nend","test_suite":"%%\r\nx = 25;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))\r\n\r\n\r\n%%\r\nx = 41;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))\r\n\r\n%%\r\nx = 7;\r\ny=sumOfSqr(x);\r\nassert(isempty(y));\r\n\r\n%%\r\nx = 391250;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))\r\n\r\n%%\r\nx = 288;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))\r\n\r\n%%\r\nx = 1;\r\ny=sumOfSqr(x);\r\nv=y(1)^2 + y(2)^2;\r\nassert(isequal(x,v))\r\nassert(round(y(1))==y(1))\r\nassert(round(y(2))==y(2))","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":17203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":42,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2014-06-18T06:53:27.000Z","updated_at":"2026-05-28T02:36:32.000Z","published_at":"2014-06-18T06:54:02.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer n, express it as a sum of two squares if possible. Return empty matrix otherwise. Solution may not be unique and uniqueness will not be checked, so return one of the solutions. Example:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[n=25\\nOutput=[3 4]\\n\\nn=41\\nOutput=[5 4]\\n\\nn=7\\nOutput=[]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":45189,"title":"Double Factorial","description":"Return double factorial\r\n\r\n        n(n-2)...(5)(3)(1),   n\u003e0, odd\r\nn!!  =  n(n-2)...(6)(4)(2),   n\u003e0, even\r\n        1                 ,   n = -1, or n = 0\r\n\r\nFor example\r\n\r\n6!! = (6)(4)(2) = 48","description_html":"\u003cp\u003eReturn double factorial\u003c/p\u003e\u003cpre\u003e        n(n-2)...(5)(3)(1),   n\u0026gt;0, odd\r\nn!!  =  n(n-2)...(6)(4)(2),   n\u0026gt;0, even\r\n        1                 ,   n = -1, or n = 0\u003c/pre\u003e\u003cp\u003eFor example\u003c/p\u003e\u003cp\u003e6!! = (6)(4)(2) = 48\u003c/p\u003e","function_template":"function y = doubleFactorial(x)\r\n  y = x;\r\nend","test_suite":"\r\ncorrectList = [1 1 2 3 8 15 48 105 384 945 3840 10395 46080 135135 645120 2027025 10321920];\r\nfor idx = 1:17\r\nassert(isequal(doubleFactorial(idx-1),correctList(idx)))\r\nend","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":8703,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":69,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2019-11-01T18:08:07.000Z","updated_at":"2026-05-30T02:33:35.000Z","published_at":"2019-11-01T18:08:34.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn double factorial\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[        n(n-2)...(5)(3)(1),   n\u003e0, odd\\nn!!  =  n(n-2)...(6)(4)(2),   n\u003e0, even\\n        1                 ,   n = -1, or n = 0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e6!! = (6)(4)(2) = 48\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60948,"title":"Check p² = 24k + 1, p \u003e 3, the 'golden prime squares' equation","description":"Historical context\r\nIn december 2023, I / Nicolas Douillet was working on prime squares properties and I found* the formula :\r\n\r\n                                                   \r\n\r\nwhich I named the ‘Christmas eve equation'. From it I then wrote :\r\n\r\n\r\n\r\nand I named it the ‘golden prime squares equation’. These two formulas, which are actually the same one, can easily be proven starting from  (note this last formula applies not only to prime numbers). \r\n \r\n* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website :  Site de Gérard Villemin\r\n \r\nProblem statement\r\nYour goal here is to check this formula for the first primes less or equal to a given integer m (input). To do so you will simply compute k in a vector. For the two first cases where the formula doesn’t apply -2 and 3- your function should return 0.\r\n \r\nExamples\r\n\r\nIf m = 20 then k = [0, 0, 1, 2, 5, 7, 12, 15], since 5^2 = 24x1 + 1, 7^2 = 24x2 + 1, 11^2 = 24x5 + 1, etc…\r\n\r\nIf m = 40 then k = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ; \r\n \r\nFobidden functions\r\n \r\nregexp\r\nstr2num\r\nassignin\r\n \r\nSee also (my file exchange contributions using this equation) :\r\n\r\nIsprime optimus [FAST]\r\nPrime number spiral (my corresponding contribution on my file exchange)\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1019.47px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 509.733px; transform-origin: 408px 509.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0583px 8px; transform-origin: 68.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn december 2023, I / \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/profile/authors/8668631?s_tid=user_nav_comm\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eNicolas Douillet\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 210.05px 8px; transform-origin: 210.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e was working on prime squares properties and I found* the formula :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"251\" height=\"19\" style=\"width: 251px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.025px 8px; transform-origin: 99.025px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                                   \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.7333px 8px; transform-origin: 58.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhich I named the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.4167px 8px; transform-origin: 76.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e‘Christmas eve equation'\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0417px 8px; transform-origin: 68.0417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. From it I then wrote :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"249\" height=\"19\" style=\"width: 249px; height: 19px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.95px 8px; transform-origin: 57.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand I named it the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.2167px 8px; transform-origin: 99.2167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e‘golden prime squares equation’\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.483px 8px; transform-origin: 215.483px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. These two formulas, which are actually the same one, can easily be proven starting from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"111.5\" height=\"18\" style=\"width: 111.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 182.033px 8px; transform-origin: 182.033px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (note this last formula applies not only to prime numbers). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 299.367px 8px; transform-origin: 299.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website : \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eSite de Gérard Villemin\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 233.5px 8px; transform-origin: 233.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYour goal here is to check this formula for the first primes less or equal to a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.4917px 8px; transform-origin: 45.4917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003egiven integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8.16667px 8px; transform-origin: 8.16667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 97.8417px 8px; transform-origin: 97.8417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(input). To do so you will simply compute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.575px 8px; transform-origin: 211.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector. For the two first cases where the formula doesn’t apply -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e3\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.9083px 8px; transform-origin: 94.9083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e- your function should return 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 20\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 82.6333px 8px; transform-origin: 82.6333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [0, 0, 1, 2, 5, 7, 12, 15]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.4333px 8px; transform-origin: 26.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.075px 8px; transform-origin: 46.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, 7^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.75px 8px; transform-origin: 15.75px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.4px 8px; transform-origin: 7.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e11\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.4333px 8px; transform-origin: 26.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, etc…\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 40\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 133.575px 8px; transform-origin: 133.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 161.05px 8px; transform-origin: 161.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(my file exchange contributions using this equation) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/171714-isprime-optimus-fast?s_tid=srchtitle\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eIsprime optimus [FAST]\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/171709-prime-numbers-spiral?s_tid=prof_contriblnk\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime number spiral (my corresponding contribution on my file exchange)\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function k = golden_prime_squares(m)\r\n\r\n  k = m;\r\n  \r\nend","test_suite":"%%\r\nm = 4;\r\nk_correct = [0, 0];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%%\r\nm = 20;\r\nk_correct = [0, 0, 1, 2, 5, 7, 12, 15];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%%\r\nm = 40;\r\nk_correct = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('golden_prime_squares.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:55:57.000Z","deleted_by":null,"deleted_at":null,"solvers_count":23,"test_suite_updated_at":"2025-07-09T05:55:18.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-27T16:53:40.000Z","updated_at":"2026-05-17T11:22:43.000Z","published_at":"2025-06-27T17:08:17.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn december 2023, I / \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/profile/authors/8668631?s_tid=user_nav_comm\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eNicolas Douillet\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e was working on prime squares properties and I found* the formula :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{p}, p \\\\geq 5 \\\\Rightarrow \\\\exists k \\\\in \\\\mathbb{N},  \\\\mathbf{p^2 = 24k + 25}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                                   \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich I named the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e‘Christmas eve equation'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. From it I then wrote :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{p}, p \\\\geq 5 \\\\Rightarrow \\\\exists k \\\\in \\\\mathbb{N}^*,  \\\\mathbf{p^2 = 24k + 1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand I named it the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e‘golden prime squares equation’\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. These two formulas, which are actually the same one, can easily be proven starting from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = 6n \\\\pm 1, p \\\\geq 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (note this last formula applies not only to prime numbers). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website : \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSite de Gérard Villemin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour goal here is to check this formula for the first primes less or equal to a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egiven integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(input). To do so you will simply compute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector. For the two first cases where the formula doesn’t apply -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e- your function should return 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 20\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [0, 0, 1, 2, 5, 7, 12, 15]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, 7^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e11\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, etc…\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 40\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/fileexchange/171714-isprime-optimus-fast?s_tid=srchtitle\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIsprime optimus [FAST]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/fileexchange/171709-prime-numbers-spiral?s_tid=prof_contriblnk\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime number spiral (my corresponding contribution on my file exchange)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1,"title":"Times 2 - START HERE","description":"Try out this test problem first.\r\n\r\nGiven the variable x as your input, multiply it by two and put the result in y.\r\n\r\nExamples:\r\n\r\n Input  x = 2\r\n Output y is 4\r\n\r\n Input  x = 17\r\n Output y is 34\r\n","description_html":"\u003cp\u003eTry out this test problem first.\u003c/p\u003e\u003cp\u003eGiven the variable x as your input, multiply it by two and put the result in y.\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cpre\u003e Input  x = 2\r\n Output y is 4\u003c/pre\u003e\u003cpre\u003e Input  x = 17\r\n Output y is 34\u003c/pre\u003e","function_template":"function y = times2(x) % Do not edit this line.\r\n\r\n  % Modify the line below so that the output y is twice the incoming value x\r\n\r\n  y = x;\r\n\r\n  % After you modify the code, press the \"Submit\" button, and you're on your way.\r\n\r\nend % Do not edit this line.","test_suite":"%%\r\nassert(isequal(times2(1),2));\r\n\r\n%%\r\nassert(isequal(times2(11),22));\r\n\r\n%%\r\nassert(isequal(times2(-3),-6));\r\n\r\n%%\r\nassert(isequal(times2(29),58));","published":true,"deleted":false,"likes_count":2305,"comments_count":151,"created_by":1,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":116365,"test_suite_updated_at":"2012-01-25T22:41:49.000Z","rescore_all_solutions":false,"group_id":2,"created_at":"2012-01-18T01:00:16.000Z","updated_at":"2026-07-20T23:23:59.000Z","published_at":"2012-01-18T01:00:16.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTry out this test problem first.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the variable x as your input, multiply it by two and put the result in y.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  x = 2\\n Output y is 4\\n\\n Input  x = 17\\n Output y is 34]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":45960,"title":"Set the Euler-Mascheroni constant with an identity using available MATLAB functions","description":"Cody Problem \u003chttps://www.mathworks.com/matlabcentral/cody/problems/42826 42826\u003e asked us to estimate the \u003chttps://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant Euler-Mascheroni constant\u003e gamma = 0.5772156649... by summing terms in an infinite series. This problem asks you to set the constant with an identity that uses functions available in MATLAB. For example, when I started programming, I read some code that set pi not with an assignment such as pi = 3.14159 but with the identity pi = 4*atan(1).\r\n\r\nFind an analogous identity and use it to set gamma to 10 decimal places. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114.9px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57.45px; transform-origin: 407px 57.45px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84.9px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42.45px; text-align: left; transform-origin: 384px 42.45px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.35px 7.8px; transform-origin: 44.35px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCody Problem\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/42826\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e42826\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.7833px 7.8px; transform-origin: 77.7833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asked us to estimate the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eEuler-Mascheroni constant\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eγ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.783px 7.8px; transform-origin: 134.783px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e= 0.5772156649... by summing terms in an infinite series. This problem asks you to set the constant with an identity that uses functions available in MATLAB. For example, when I started programming, I read some code that set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 101.517px 7.8px; transform-origin: 101.517px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e not with an assignment such as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.2px 7.8px; transform-origin: 46.2px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 46.2px 8.25px; transform-origin: 46.2px 8.25px; \"\u003epi = 3.14159\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.8px; transform-origin: 28px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e but with the identity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.9px 7.8px; transform-origin: 53.9px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 53.9px 8.25px; transform-origin: 53.9px 8.25px; \"\u003epi = 4*atan(1)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.983px 7.8px; transform-origin: 134.983px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFind an analogous identity and use it to set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eγ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.6833px 7.8px; transform-origin: 67.6833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to 10 decimal places.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function gamma = EulerMascheroni\r\n  gamma = ...;\r\nend","test_suite":"%%\r\ny_correct = 0.5772156649;\r\nassert(isequal(EulerMascheroni,y_correct))\r\n\r\n%%\r\nfiletext = fileread('EulerMascheroni.m');\r\nsumming = ~isempty(strfind(filetext, 'sum')) || ~isempty(strfind(filetext, 'plus')) ||...\r\n          ~isempty(strfind(filetext, '+'));\r\nassert(~summing, 'Please do not sum infinite series')\r\n\r\n%%\r\nfiletext = fileread('EulerMascheroni.m');\r\nassert(isempty(strfind(filetext, '5772156649')), 'Please do not set the constant directly (or indirectly with arithmetic')\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-16T05:45:50.000Z","updated_at":"2026-07-14T07:59:58.000Z","published_at":"2020-06-16T05:53:07.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/42826\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e42826\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asked us to estimate the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eEuler-Mascheroni constant\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"gamma\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\gamma\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e= 0.5772156649... by summing terms in an infinite series. This problem asks you to set the constant with an identity that uses functions available in MATLAB. For example, when I started programming, I read some code that set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e not with an assignment such as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epi = 3.14159\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e but with the identity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epi = 4*atan(1)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind an analogous identity and use it to set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"gamma\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\gamma\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to 10 decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":61420,"title":"Sum of proper divisors (aliquot sum)","description":"A proper divisor of a positive integer n is any divisor of n except n itself. The aliquot sum is the sum of all proper divisors.\r\n\r\nGiven a positive integer n, return s, the sum of its proper divisors.\r\n\r\nFor example, the proper divisors of 12 are 1, 2, 3, 4, and 6, so the aliquot sum is 16.\r\n\r\nNote: the aliquot sum of 1 is 0 (it has no proper divisors).\r\n\r\nExamples:\r\n  n = 12  -\u003e  16   (1 + 2 + 3 + 4 + 6)\r\n  n = 6   -\u003e  6    (1 + 2 + 3, so 6 is a \"perfect number\")\r\n  n = 1   -\u003e  0\r\n  n = 7   -\u003e  1    (prime, only proper divisor is 1)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 423px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 244px 211.5px; transform-origin: 244px 211.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 21px; text-align: left; transform-origin: 220px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA proper divisor of a positive integer n is any divisor of n except n itself. The aliquot sum is the sum of all proper divisors.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a positive integer n, return s, the sum of its proper divisors.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 21px; text-align: left; transform-origin: 220px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, the proper divisors of 12 are 1, 2, 3, 4, and 6, so the aliquot sum is 16.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNote: the aliquot sum of 1 is 0 (it has no proper divisors).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eExamples:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  n = 12  -\u0026gt;  16   (1 + 2 + 3 + 4 + 6)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  n = 6   -\u0026gt;  6    (1 + 2 + 3, so 6 is a \"perfect number\")\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  n = 1   -\u0026gt;  0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 10.5px; text-align: left; transform-origin: 220px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  n = 7   -\u0026gt;  1    (prime, only proper divisor is 1)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = aliquot(n)\r\n    s = n;\r\nend","test_suite":"%%\r\nassert(isequal(aliquot(12), 16))\r\n%%\r\nassert(isequal(aliquot(6), 6))\r\n%%\r\nassert(isequal(aliquot(1), 0))\r\n%%\r\nassert(isequal(aliquot(7), 1))\r\n%%\r\nassert(isequal(aliquot(28), 28))   % 1+2+4+7+14, another perfect number\r\n%%\r\nassert(isequal(aliquot(100), 117))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":5142412,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2026-07-08T14:10:26.000Z","updated_at":"2026-07-18T00:22:34.000Z","published_at":"2026-07-08T14:10:26.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA proper divisor of a positive integer n is any divisor of n except n itself. The aliquot sum is the sum of all proper divisors.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a positive integer n, return s, the sum of its proper divisors.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, the proper divisors of 12 are 1, 2, 3, 4, and 6, so the aliquot sum is 16.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNote: the aliquot sum of 1 is 0 (it has no proper divisors).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e  n = 12  -\u0026gt;  16   (1 + 2 + 3 + 4 + 6)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e  n = 6   -\u0026gt;  6    (1 + 2 + 3, so 6 is a \\\"perfect number\\\")\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e  n = 1   -\u0026gt;  0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e  n = 7   -\u0026gt;  1    (prime, only proper divisor is 1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":45224,"title":"Wythoff Sequence","description":"\r\nFind the lower Wythoff sequence up to n.\r\n\r\nFor more information, \u003chttps://oeis.org/A000201\u003e","description_html":"\u003cp\u003eFind the lower Wythoff sequence up to n.\u003c/p\u003e\u003cp\u003eFor more information, \u003ca href = \"https://oeis.org/A000201\"\u003ehttps://oeis.org/A000201\u003c/a\u003e\u003c/p\u003e","function_template":"function y=wythoff(n)\r\n  y = x;\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = 1;\r\nassert(isequal(wythoff(n),y_correct))\r\n%%\r\nn = 10;\r\ny_correct = [1,3,4,6,8,9,11,12,14,16];\r\nassert(isequal(wythoff(n),y_correct))\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":363598,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2019-12-04T12:02:31.000Z","updated_at":"2026-05-30T02:36:18.000Z","published_at":"2019-12-04T12:20:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the lower Wythoff sequence up to n.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor more information,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://oeis.org/A000201\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;https://oeis.org/A000201\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":55275,"title":"List the semiprimes","description":"A semiprime number—or a 2-almost prime—is the product of two prime numbers. The numbers  and  are semiprimes, but  and  are not. The semiprimes appear in three of Ramon Villamangca’s “Easy Sequences”: Cody Problems 52859, 52990, and 53740. \r\nWrite a function to list the semiprime numbers less than or equal to the input number. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297.167px 8px; transform-origin: 297.167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA semiprime number—or a 2-almost prime—is the product of two prime numbers. The numbers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHkAAAAkCAYAAACt6pVgAAAElUlEQVR4Xu1auapUQRB97wPELTUQl8BIwSXSQMEFDVVUNBAE19w1dRcMDNx+QEXNVFxAQUVwCTQycEEMjFzQH9BzoEua+3qp7rkzc/tNXzjMnbm9VNXpqq7qO+Nj9Zr0Fhif9BpWBccqySOwCCrJleQRsMAIqFg9uZLstMBC/LoWmG2e3sHnvcJsRR2mGJmfFyY7xV1uZH6Pzx8x+bWePBMD7QJ2A3OBQ8At4HNsgo49Pwh5DgPTG3KdxPfzGoMNUR8uzCPAloYMD/B9f4gLDckc/LYhNzrgEI0Qm/o1GiwJNKJu2ztKNDl4AvwErhodNlv6/ML9fJ/sMZK3ouM1M2gJq93HIT34DLAXYARiiJsDbAJsz6aOx2KrZcDPGUU/APTiK425bX4YXc+6ZAuRzLj/zHTq8iqP2ZxG+g5sA647Gtt60iNmxAYc8HMu0N8OgkWMj7jhFnoDIOkTLh/Jsnq4d1FxhrnS9l9Rdg9uVvoMYBrZoXwRfnunIJI2ug88BFK8X0LvDvTTJKyUv+nBtniUgYmwNwr5SL6ITvvMSF0MYQoO/jehJ7wAQlm0hHN20pLc3Mo0RAvBdJ62oqN4slduF8kS3sRKDAX0Yik7vhXs1b7FISQHExhH5xSi+0GwbDVBR3SRzPBw2Sj0CZ/M5pplB38/FwkjLoNyAS1IcUNP27YXmkQu774WkFlDdD8IljFPQzZnwiUyu0hmcmLXYpfwnQcef4BZwHGA3s0rNZTbSU4vXDNLDu1TqWNLyFuBjjmHIyGi2yZYqgJWC3IFKx8XyXYS4jKmJBxSc27ATJoEggJRYVu4VDKk/YWEOWNzyMLL8WJ7bBfRbRPMOXYC5KBZ83v3eBfJf43koax6PdrcNe04+LqYJTv8nNnpPGAN0GsFYRPNCMjvbSZZthnlFNJ2GmeEi5EcqhntxeA9bekwuRSN+ccpgCWWpmzSqGMTzfZtZdG+uaMO5yKZR2dSH4dIlvqMk2vLDo2RBtWGofQtkLLdaGSTcaVtat6imaPZRrZY52GOi2QhL1ZO5JDclew69UBCa3h7D2YFkpugaueTdnYJOMExXSSfQM+jpnfIQ7WLwRa4C9n1IAiWEL0ayttn/5oDk1SC2V5IdiaPLpLtcOM77+XAUna8wf1SpWTDzq5ZflDe0JGiRJuUUiqURWvqaKX5vM2kznduPb5jTYnxPgLtU7HQQuhV+Db7S+l3E4OGDg9osKeA62WGSx5NmdRPouU9wysI53xV6iPZDqsuEiU8lFI+CcHcI3lC5LtW4cEyQFstaAiWuXKJFodzvcsXvZgobwScFULoVSOF4qrmACT6EcD3sHLsyfh/wPzWpse1PVbz8CY2vjYbTiG4F6JlW5QxWH9/AaYCPJkkD8F/tcT+NEAD8cU668hpZpbH+Iy91YkZcpDPGXXoodqLL981NbN4ZmodnNqPeQT/erXYUoAcfAXE8YK6xUjWGmZU2/Eg4iUQ/TNdw0C5/bLsXEnOMltZnSrJZfGVJW0lOctsZXWqJJfFV5a0leQss5XVqZJcFl9Z0laSs8xWVqdKcll8ZUn7D3PYHzR1iFOGAAAAAElFTkSuQmCC\" alt=\"6 = 2x3\" style=\"width: 60.5px; height: 18px;\" width=\"60.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"9 = 3x3\" style=\"width: 60.5px; height: 18px;\" width=\"60.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are semiprimes, but \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKgAAAAkCAYAAAAdOXEXAAAFeUlEQVR4Xu1aN8sWTRR93x9g9geIobBSMDVaKBhQsBJUtLAQTGAlKoqdfsHCwsIAloKKtopaKKgIRrSyMJRWhp+g58BeGJYJ984G58FZuDxhd3bunDlz08z0VL0qAgUjMF2wblW1isBUJWglQdEIVIIWPT1VuUrQyoGiEagELXp6qnJagi4DVDMauD7g8/uEQ+eO59mEjWUe9F3a6PwVn18mTH8T9jGCLsTAj0F2Qf5rQFjQ/H6Jz8MTCM5x6HwSMqc1qf/g94XCFx4n9n/I5pbun/F7H6T0hZaFfYigBOMxhETc05o4uUecVk4QSV81+oYMzgPPWEsxTluhyN2EMttw/14pCrf0yMY+RNBP6GARJDToA7h3BXIZQkta+sXVS+tzEHKnWXD0EDsgrkWlJT1d2GDo0j9CaCzOQsRSkrRHIa5Fnd+MraQhdMLeR1BayHfNCJfj871ntGvx31PIT8jcktDw6MIJ/gbZDbkZGQtvlTiec9BrE2RLgHz3HZJyAV4taD46Y+8jqJCP4wxZUHmG8c/iggDxqUJrvx7CWDp0uS4otCjbbQk+yfEQYrG6EiLtRTuNS/7R6O8zFNTJdf+3EuN0x0A8LkGsYRrb0Pu0Q79BsPcRVFjPDl9DfCuXq/oUpESX2AaKLuY5JJZEiBtiWy1BOcE3ms60OAg5maRpYl5JVFNh1K9GDy1BZXGRnPQaWpKSnIeavjQxb2fsQzGoq0gbSAGZK5uuZ9LKHL6VLgTlZC2BaMtoFpJayenTM/SfEPQEHjivbGglqZWcSjWmotiHCOoqz444cdshMyHXIYzlzhgmUpR1a3jaAfie67v+J+BrLZCrk4akQ5KTVpahFi+t9Xfng2FKypIORU7qEcU+Vgdtk1SIqo2dfMRy49suBO07GZCqxToolVNPjJF0SHISQ+mbno7hmPVKWdIhyUldo9indpJCJGXSEQraYwBJsdkKYvv5i/hDk2Bo+pFFk2M9U5Z0aHKyf8nirdbT1T1E0qHJmcQ+RVCJD5gEMDh2d2A0QbKGIH/6GU4wKxF9xNOuJWWNmL+1CVEODjLBltgz1E+bpAzjLAlRjv5J7GMEldUjRGSsw//cwnCuS8wZzBBtWIL6F5LrEXw6uSTlfU22njM2EuoFhGWuVJavfb/PYw5liFTYhwgqltPn9lyzPwl10NDkyIZE3xPgbnSwb20JSksieY4WbjZEU4+0vJvv3dk0sJSgLH2osQ8RlCUkuqbQ5LmDsFjRUrJ4a7FcC74bc3Lxcrt4CJJaiuVa3flc2/hQ/75JasI+tZMUCrxdK2HJqEvI4k0AGWbXlxBtRHtrMT/V5RjkpGFi+KApQaX0de+bsc8lKDuV4rCFoH86i2cczd2xWKlMrLyl3BTL1jV1Uu0kcwdvRcKt0whYz+yGsvVUCUqrN5/Lwj7l4kMHLNztULqBSdhNErBvQ9/Ybgsn6wnEd7DENyGaUlIfJOU7eJopVm2gLtcgqwzMSZWS+iBpNvapJCm0Fy/H7brWDg04dnpUAOJiksPXvhduwJ+rIdrtTg05pZ8uJJW2xPttAIlZ+J9lIY5Pu92ZIqd01YWknbCPlZnkQAhJuh8ihXkpD7C8cQSi3bfuxLAOjX2lk9jrtFm3hZxdSNouW8V0tyQ0WnJ2IWln7FOFejnUS8si1yN8SZ0O6sCn3puyZObqn+qARW/NLpm7xWgp9VjaSe05pbPcf4MvmqN/LnEsZTZru87YpwiqBeZvfY5nMZntWr1Ibrs+cSbZ1kCsW8a57bJ0rwTNgq02GguBStCxkK79ZCFQCZoFW200FgKVoGMhXfvJQqASNAu22mgsBCpBx0K69pOFQCVoFmy10VgIVIKOhXTtJwuB38VvnzR/zZ/+AAAAAElFTkSuQmCC\" alt=\"8 = 2x2x2\" style=\"width: 84px; height: 18px;\" width=\"84\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"30 = 2x3x5\" style=\"width: 91.5px; height: 18px;\" width=\"91.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 205.517px 8px; transform-origin: 205.517px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are not. The semiprimes appear in three of Ramon Villamangca’s “Easy Sequences”: Cody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/52859\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e52859\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/52990\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e52990\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/53740\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e53740\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 264.75px 8px; transform-origin: 264.75px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to list the semiprime numbers less than or equal to the input number. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = semiprimes(n)\r\n  s = primes(n)/2;\r\nend","test_suite":"%%\r\nn = 100; \r\ns = semiprimes(n);\r\ns_correct = [4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 58 62 65 69 74 77 82 85 86 87 91 93 94 95];\r\nassert(isequal(s,s_correct))\r\n\r\n%%\r\nn = 1000; \r\ns = semiprimes(n);\r\ns_correct = [4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 58 62 65 69 74 77 82 85 86 87 91 93 94 95 106 111 115 118 119 121 122 123 129 133 134 141 142 143 145 146 155 158 159 161 166 169 177 178 183 185 187 194 201 202 203 205 206 209 213 214 215 217 218 219 221 226 235 237 247 249 253 254 259 262 265 267 274 278 287 289 291 295 298 299 301 302 303 305 309 314 319 321 323 326 327 329 334 335 339 341 346 355 358 361 362 365 371 377 381 382 386 391 393 394 395 398 403 407 411 413 415 417 422 427 437 445 446 447 451 453 454 458 466 469 471 473 478 481 482 485 489 493 497 501 502 505 511 514 515 517 519 526 527 529 533 535 537 538 542 543 545 551 553 554 559 562 565 566 573 579 581 583 586 589 591 597 611 614 622 623 626 629 633 634 635 649 655 662 667 669 671 674 679 681 685 687 689 694 695 697 698 699 703 706 707 713 717 718 721 723 731 734 737 745 746 749 753 755 758 763 766 767 771 778 779 781 785 789 791 793 794 799 802 803 807 813 815 817 818 831 835 838 841 842 843 849 851 862 865 866 869 871 878 879 886 889 893 895 898 899 901 905 913 914 917 921 922 923 926 933 934 939 943 949 951 955 958 959 961 965 973 974 979 982 985 989 993 995 998];\r\nassert(isequal(s,s_correct))\r\n\r\n%%\r\nn = 10000; \r\ns = semiprimes(n);\r\nlen_correct = 2625;\r\nsum_correct = 12736914;\r\nvar_correct = 8.447173943104530e+06;\r\nassert(isequal(length(s),len_correct) \u0026\u0026 isequal(sum(s),sum_correct) \u0026\u0026 abs(var(s)-var_correct)\u003c1e-8)\r\n\r\n%%\r\nn = 100000; \r\ns = semiprimes(n);\r\nlen_correct = 23378;\r\nsum_correct = 1138479765;\r\nvar_correct = 8.471797671132822e+08;\r\nassert(isequal(length(s),len_correct) \u0026\u0026 isequal(sum(s),sum_correct) \u0026\u0026 abs(var(s)-var_correct)\u003c1e-6)\r\n\r\n%%\r\nn = 800000; \r\ns = semiprimes(n);\r\nlen_correct = 169660;\r\nsum_correct = 66262251604;\r\nvar_correct = 5.417425253731966e+10;\r\nassert(isequal(length(s),len_correct) \u0026\u0026 isequal(sum(s),sum_correct) \u0026\u0026 abs(var(s)-var_correct)\u003c1e-4)\r\n\r\n%%\r\nfiletext = fileread('semiprimes.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'switch') || contains(filetext,'regexp'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-08-02T01:42:35.000Z","deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-08-02T01:34:03.000Z","updated_at":"2026-05-31T02:07:22.000Z","published_at":"2022-08-02T01:42:35.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA semiprime number—or a 2-almost prime—is the product of two prime numbers. The numbers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"6 = 2x3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e6 = 2\\\\times 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"9 = 3x3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e9 = 3 \\\\times 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are semiprimes, but \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"8 = 2x2x2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e8 = 2\\\\times2 \\\\times 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"30 = 2x3x5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e30 = 2\\\\times 3 \\\\times 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are not. The semiprimes appear in three of Ramon Villamangca’s “Easy Sequences”: Cody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/52859\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e52859\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/52990\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e52990\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/53740\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e53740\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to list the semiprime numbers less than or equal to the input number. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51570,"title":"Count numbers whose reciprocals' decimal expansions terminate","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 63px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 31.5px; transform-origin: 407px 31.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 378.35px 7.91667px; transform-origin: 378.35px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSome numbers (such as 4) have reciprocals with terminating decimal expansions, while others (such as 7) do not. Write a function that counts the numbers less than or equal to the input number whose reciprocals have terminating decimal expansions. For example, given 10, your function should return a count of 6, which corresponds to 1, 2, 4, 5, 8, and 10. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.1083px 7.91667px; transform-origin: 14.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e         \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = terminatingRecip(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 10;\r\ny_correct = 6;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 127;\r\ny_correct = 16;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 39841;\r\ny_correct = 61;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 63521;\r\ny_correct = 66;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 52123456;\r\ny_correct = 161;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 69871234;\r\ny_correct = 166;\r\nassert(isequal(terminatingRecip(n),y_correct))\r\n\r\n%%\r\nn = 4031321321987654;\r\ny_correct = 616;\r\nassert(isequal(terminatingRecip(n),y_correct))","published":true,"deleted":false,"likes_count":5,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-27T03:16:08.000Z","updated_at":"2026-07-15T12:23:16.000Z","published_at":"2021-04-27T03:19:01.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSome numbers (such as 4) have reciprocals with terminating decimal expansions, while others (such as 7) do not. Write a function that counts the numbers less than or equal to the input number whose reciprocals have terminating decimal expansions. For example, given 10, your function should return a count of 6, which corresponds to 1, 2, 4, 5, 8, and 10. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e         \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":47803,"title":"Check if the input is a strong prime","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 109.995px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.491px 54.9977px; transform-origin: 406.497px 54.9977px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20.5882px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10.2941px; text-align: left; transform-origin: 383.496px 10.2941px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMore information about strong primes on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Strong_prime\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eWikipedia\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 79.4118px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 403.493px 39.7059px; transform-origin: 403.498px 39.7059px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisStrongPrime(11) = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisStrongPrime(2) = 0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisStrongPrime(9) = 0; \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisStrongPrime(227) = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = isStrongPrime(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 2;\r\ny_correct = 0;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nstrongPrimeList = [11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499];\r\nfor idx = 1:100\r\n    x = randi([2 500]);\r\n    y_correct = 0;\r\n    if ismember(x, strongPrimeList)\r\n        y_correct = 1;\r\n    end\r\n    assert(isequal(isStrongPrime(x),y_correct))\r\nend\r\n\r\n%%\r\nx = 4673;\r\ny_correct = 1;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 4677;\r\ny_correct = 0;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 46747;\r\ny_correct = 1;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 46748;\r\ny_correct = 0;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 467317;\r\ny_correct = 1;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n\r\n%%\r\nx = 467318;\r\ny_correct = 0;\r\nassert(isequal(isStrongPrime(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":8703,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":30,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-04T16:46:35.000Z","updated_at":"2026-05-30T19:13:47.000Z","published_at":"2020-12-04T16:46:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMore information about strong primes on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Strong_prime\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eWikipedia\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[isStrongPrime(11) = 1;\\nisStrongPrime(2) = 0;\\nisStrongPrime(9) = 0; \\nisStrongPrime(227) = 1;]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57869,"title":"Identify de Polignac numbers","description":"The numbers 125 and 329 can be written as the sum of a prime and a power of 2. For example, , and . The numbers 127 and 331, which are examples of de Polignac numbers, cannot be written in this way.\r\nWrite a function that determines whether an odd number is a de Polignac number. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 36px; transform-origin: 407px 36px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297.55px 8px; transform-origin: 297.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe numbers 125 and 329 can be written as the sum of a prime and a power of 2. For example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"125 = 109+2^4\" style=\"width: 96px; height: 19px;\" width=\"96\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"329 = 73+2^8\" style=\"width: 88.5px; height: 19px;\" width=\"88.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 320.392px 8px; transform-origin: 320.392px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The numbers 127 and 331, which are examples of de Polignac numbers, cannot be written in this way.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 255.042px 8px; transform-origin: 255.042px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that determines whether an odd number is a de Polignac number. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = isdePolignac(n)\r\n  tf = isequal(n,p+2^k);\r\nend","test_suite":"%%\r\nassert(isdePolignac(1))\r\n\r\n%%\r\nassert(~isdePolignac(17))\r\n\r\n%%\r\nassert(~isdePolignac(75))\r\n\r\n%%\r\nassert(isdePolignac(127))\r\n\r\n%%\r\nassert(isdePolignac(331))\r\n\r\n%%\r\nassert(~isdePolignac(531))\r\n\r\n%%\r\nassert(isdePolignac(905))\r\n\r\n%%\r\nassert(isdePolignac(1619))\r\n\r\n%%\r\nassert(~isdePolignac(2261))\r\n\r\n%%\r\nassert(isdePolignac(7535))\r\n\r\n%%\r\nassert(~isdePolignac(10413))\r\n\r\n%%\r\nassert(isdePolignac(21453))\r\n\r\n%%\r\nassert(isdePolignac(45233))\r\n\r\n%%\r\nassert(~isdePolignac(70999))\r\n\r\n%%\r\nassert(~isdePolignac(96415))\r\n\r\n%%\r\nassert(~isdePolignac(121399))\r\n\r\n%%\r\nassert(isdePolignac(148243))\r\n\r\n%%\r\nassert(isdePolignac(172841))\r\n\r\n%%\r\nassert(isdePolignac(201599))\r\n\r\n%%\r\nassert(isdePolignac(227107))\r\n\r\n%%\r\nassert(isdePolignac(253151))\r\n\r\n%%\r\nassert(~isdePolignac(267267))\r\n\r\n%%\r\nassert(~isdePolignac(271271))\r\n\r\n%%\r\nassert(isdePolignac(273421))\r\n\r\n%%\r\nassert(isdePolignac(542459))\r\n\r\n%%\r\nassert(isdePolignac(2000039))\r\n\r\n%%\r\nassert(~isdePolignac(123456789))\r\n\r\n%%\r\nassert(isdePolignac(123456791))\r\n\r\n%%\r\nassert(isequal(bin2dec(num2str(arrayfun(@isdePolignac,5893:2:5933)')'),288))\r\n\r\n%%\r\nassert(isequal(bin2dec(num2str(arrayfun(@isdePolignac,21671:10:21791)')'),4624))\r\n\r\n%%\r\nfiletext = fileread('isdePolignac.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2023-03-29T05:27:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":24,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-03-29T05:27:08.000Z","updated_at":"2026-05-25T01:35:42.000Z","published_at":"2023-03-29T05:27:20.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe numbers 125 and 329 can be written as the sum of a prime and a power of 2. For example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"125 = 109+2^4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e125 = 109+2^4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"329 = 73+2^8\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e329 = 73+2^8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The numbers 127 and 331, which are examples of de Polignac numbers, cannot be written in this way.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that determines whether an odd number is a de Polignac number. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60782,"title":"Convert integers from primary notation to base 10 ","description":"The primary system of expressing numbers has three digits: m, x, and p, which correspond to -1, 0, and 1. The places in a primary number correspond to the prime numbers. For example, the number 6 could be ppm in primary notation because . Similarly, the number 20 could be ppxxp in primary notation. \r\nBecause integers can be expressed as sums (and differences) of primes in several ways, several primary notations for a number can exist. Cody Problem 60783 deals with proper primary notation.  \r\nWrite a function to convert integers expressed in primary notation to base 10. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 145.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 72.75px; transform-origin: 407px 72.75px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 64.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32.25px; text-align: left; transform-origin: 384px 32.25px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.5px 8px; transform-origin: 14.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.5px 8px; transform-origin: 25.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eprimary \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 150px 8px; transform-origin: 150px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esystem of expressing numbers has three digits: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003ex\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160px 8px; transform-origin: 160px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which correspond to -1, 0, and 1. The places in a primary number correspond to the prime numbers. For example, the number 6 could be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12px 8px; transform-origin: 12px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 12px 8.5px; transform-origin: 12px 8.5px; \"\u003eppm\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91.5px 8px; transform-origin: 91.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in primary notation because \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUYAAAAlCAYAAAA+wECwAAAIV0lEQVR4Xu2dTeh3QxTHPXsSViwoLIiw8BaxkCgloRCSIi8LWcgTFpKEsJGFl1KS95KUlJcsSHkthVigSJ6Vt9hzPnVPzXP73Tuvd2bu7fzq9Dz9f3Pnnvmeme+cOWdmfnsOsI8hYAgYAobAfgjsMTwMAUPAEDAE9kfAiNF6hCFgCBgCIwSMGK1LGAKGgCFgxGh9wBAwBAoicJrUdbTIqwXrLFnVBVLZnyKfx1RqHmMMWlbWEDAEXAQgxSdELhL5o1NoDhW9XhZ5Q+SpUB1TiREWZpZAvux4tgjFQcv1PvvFtsfKGwJLIcBYeV7k7I5JUdsOOX4s8ngoOcYQI2R4wzA7fCv/PifynsiPSyG/UL24/FdM1P2v/P2Uhm26Rd59r8glIlGu/0JYhVaL3peKnCVyoMg+kY9Ebu1g0DAo7hK5VuSI0AZ1UE71vlh0OW7Qpxdc0e1nkctF3m2EVaxdKf9N6NgKIUZ1RSFGDHO/SLBL2gi0qdceI1/8MKMTRr6wgc5KiIcP7z5d/l0LMepE873o/LfICQM50pSWE40OHMgZskaXgxrYNuWVeGNvimh/GNfREld0+WQgxitTGpf5TI5d75Z3XydyvE8HHzG6BvpUKrttRQN2V9sZxCeJ3D4BzBfy95qxEvC9cdDlaodQ1kKMT4vO5w54up4DHfDBoV2vyb+1BxDvu0zkL5GbBj3WRIzfic4Hi7wg8v6gP96Z20datQcH6R2RY0VqrxZL2PW3Ade9c+Q4R4x4V18NgxVP8ZwGQMzpHvudeot4EL14vOiknesR+f+dQ6OWIkZmej7XFLAlur8lMhVjYvAwiFoMYBdX2nzGgnpoAgJvucRqg0kFB2RXOEWXg+pJtujLLb3FEnZlnIHb7OphjhiZtYht0LFbxhJiCXCqvBLPYVKgplcYqn8NYvyvIPEygEm8TcWYtD1Mqi1je0sTo3pQpSYAPJq5cBVhlycHO9b2xpkEPhO5SqT19pxUuwY5SFPE6A7SVnG3UEIJKafBYmJNfDSI/ezMwA6pt2SZtRGjr+0ss1nGPiNys6/wgt+nDqBQlUoSI4OWpNXcREJf/n1QrvbYVJv24Fzk2JXJ52uRSQ9/ihj/kYeURHi4VeYptHP6yrmkMy4LSfaQBd4aMdL5+LQOweQMIF+/4vuSxBjyPsqo51970mEVycebvAhtSEa5HLsS5mEHxeRyehcxuq66LqPvGCpyt2I8Jn9bS+aUJcD5IoeIkCwg5jT+tF4ebIkYWWYdJdJDsi5nAIWM29rE6HqMtfsshEwS9swQYBYuk2NXHWuTCaRdxOju84MYfxX5UITgMp3d3QMYaxjtRLmY3SMVPJRRyTjlr1W1yLTpu7dAjCwFXxwmHvoOsbDZ7F+GDUMfzRlAIe+oTYxkZl8RqR271fjiox3YFLvk2FXH2uRqeBcx6gt5OcHd8SZdAPpARJfaMRnUXohRO7xmVnUDbe1gtjvw1kyM2JWEwS5PnP2NLZdeOQOoR2JUxyXWKQlpy1wZHbtbIEZty2RWfxcxavyCGf88kV3L5ZaZsVwDj593EzMpmUWwYIme48Gi05qJ0cVUT0i5K4uUWFgpXLdEjJpRTU260NdJhP0kEptV3iIxTpK8jxingpNunCOFTEqTW2597obkGA+4ZPa+BDG63n4qJqXsqUs+9IitsySuJYhRnYVUTHkulczcd+o+1NRLG9z+ERuOiiHGEv3QNw5z7OptSyoxYizd58j/fY3I6VA1nnWPCsa0RTcxo2NuzGdrxAgmbry6Fa45A0j7Xg/ESP/gvHfODgp3t0ls2MhLJs5A7Z0YddKeDEfMxRh9s7zb+JhOX4PoUt6RsvlZj0zyvtwz5CWI0dfulDb66pz73vUaY/pISVxLEONcG2skXwgrlLhchHpYPpJQ5cxwzK4SdR62EGNMSr6EzvJuhwu9kaa35Ivb4SEN32SQQxK+Z7dIjC5psKOhxYmjtRNjKVL09b+Q77e2XWdyo/ouj9Gd5ecyX7qUjllC9kqM2ubY5UVIZwots0ViVFxLxNdCcRyXWzMxhpAi8f5aEw5YHinS8oin2jfHrt52TJ184dQCB9XnOrQuy3pxrVMHDs8BFNdltTwTvkViZPVBooAbhGKzoDn2dJ/NGUAhOiy1lNZwwvXDONyli17AWuuyWE1Srv1IILHWt0Umb32aIkZflla36+AtnlhxxgrpqOMySjgsk7kKa7ytRtvSmuDXRozgxqDlwtJdF9L2gusaiVH3CjN4Oc+/68MWsftEOHxR6yx60AUMKYM04ZlUu+pENnvUee52HY01ji/FVKPBujkZsgQskh4Z39jNhmM6FD+Qw43knOXNTZwkKTZ6yE1mLbV5t2Tyxc3I6ykX9+5Abp7m90By93fmYDu+PGQJT6e0xzg+QDHX/hY3X+mWoZbHAnPsCh8w5mfDAb6LajG6npMmk8WxQD7MUg+L1Ipt5A4Ovdpe77GjQ/HzDLSDG0NqX7ip7WEG5sgcJDK+rZl4J/dhliSWksTIAH5AhD6iHyadX0T44SH0b9U/dJM5y3g9oYWO2P0lEU5ulVralyRG9w7UkD7f4lSRtrfF8dlcuyqhekM7PmIMMY6VWQ8COtv3cLnDelCb11QvqqVUSy+qJp44EyevsL3Behsx1uxO9i5DYBsIRP/qXgfNxtt8XSRo25gRYwcWMxUMgRUiEPWre43bF62rEWNji9nrDYEVIwDhkDnvOTSTpKMR44p7paluCHSAAMRzqkivt/yzhI7+9U8jxg56lqlgCBgCfSFgxNiXPUwbQ8AQ6AABI8YOjGAqGAKGQF8IGDH2ZQ/TxhAwBDpAwIixAyOYCoaAIdAXAv8DTHJQRMY+X3gAAAAASUVORK5CYII=\" width=\"163\" height=\"18.5\" alt=\"6 = 5*1 + 3*1 + 2*(-1)\" style=\"width: 163px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.5px 8px; transform-origin: 110.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Similarly, the number 20 could be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20px 8px; transform-origin: 20px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 20px 8.5px; transform-origin: 20px 8.5px; \"\u003eppxxp\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.5px 8px; transform-origin: 64.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in primary notation. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380px 8px; transform-origin: 380px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBecause integers can be expressed as sums (and differences) of primes in several ways, several primary notations for a number can exist. \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/60783\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 60783\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35px 8px; transform-origin: 35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eproper\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59px 8px; transform-origin: 59px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e primary notation.  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 245px 8px; transform-origin: 245px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to convert integers expressed in primary notation to base 10. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = primary2dec(s)\r\n  y = base2dec(s,'primes')\r\nend","test_suite":"%%\r\nassert(isequal(primary2dec('ppm'),6))\r\n\r\n%%\r\nassert(isequal(primary2dec('ppxxp'),20))\r\n\r\n%%\r\nassert(isequal(primary2dec('x'),0))\r\n\r\n%%\r\nassert(isequal(primary2dec('pm'),1))\r\n\r\n%%\r\nassert(isequal(primary2dec('mp'),-1))\r\n\r\n%%\r\nassert(isequal(primary2dec('pxxpp'),16))\r\n\r\n%%\r\nassert(isequal(primary2dec('pxppxp'),27))\r\n\r\n%%\r\nassert(isequal(primary2dec('ppppp'),28))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmmmpm'),-35))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmxmmxmm'),-59))\r\n\r\n%%\r\nassert(isequal(primary2dec('pppppppm'),73))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmmmxmmmmm'),-116))\r\n\r\n%%\r\nassert(isequal(primary2dec('ppppppppppppm'),234))\r\n\r\n%%\r\nassert(isequal(primary2dec('pppppxppppppppppppxp'),589))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmmmmmmmxmmmmmmmmmmmmmmmmmmmxmmx'),-1753))\r\n\r\n%%\r\nassert(isequal(primary2dec('pppppppxpppppppppppppppppppppppppppppppppppppppxppppp'),5618))\r\n\r\n%%\r\nassert(isequal(primary2dec('mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmxmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmxmxm'),-74921))\r\n\r\n%%\r\nassert(isequal(primary2dec('ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppxpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppxpp'),514888))\r\n\r\n%%\r\nfiletext = fileread('primary2dec.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'str2num'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2025-01-06T21:00:45.000Z","deleted_by":null,"deleted_at":null,"solvers_count":11,"test_suite_updated_at":"2025-01-02T00:40:47.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-12-31T17:58:19.000Z","updated_at":"2026-06-06T01:59:11.000Z","published_at":"2024-12-31T17:58:24.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprimary \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esystem of expressing numbers has three digits: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, which correspond to -1, 0, and 1. The places in a primary number correspond to the prime numbers. For example, the number 6 could be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eppm\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in primary notation because \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"6 = 5*1 + 3*1 + 2*(-1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e6 = 5\\\\cdot 1 + 3\\\\cdot 1 + 2\\\\cdot(-1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Similarly, the number 20 could be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eppxxp\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in primary notation. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBecause integers can be expressed as sums (and differences) of primes in several ways, several primary notations for a number can exist. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/60783\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 60783\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eproper\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e primary notation.  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to convert integers expressed in primary notation to base 10. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52060,"title":"Identify Ruth-Aaron numbers","description":"After Hank Aaron hit his 715th home run and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers ( and ) comprises all of the prime numbers up to 17. \r\nHe told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*\r\nWrite a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.\r\n*The story continues in a Numberphile video on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 237px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 118.5px; transform-origin: 407px 118.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.7167px 7.91667px; transform-origin: 16.7167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAfter \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.youtube.com/watch?v=QjqYThEVoSQ\"\u003e\u003cspan style=\"perspective-origin: 70.8px 7.91667px; transform-origin: 70.8px 7.91667px; \"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eHank Aaron hit his 715\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"perspective-origin: 5.83333px 7.91667px; transform-origin: 5.83333px 7.91667px; \"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eth\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"perspective-origin: 31.5083px 7.91667px; transform-origin: 31.5083px 7.91667px; \"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003e home run\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 259.142px 7.91667px; transform-origin: 259.142px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"714 = 2*3*7*17\" style=\"width: 114.5px; height: 18px;\" width=\"114.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"715 = 5*11*13\" style=\"width: 103.5px; height: 18px;\" width=\"103.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 144.3px 7.91667px; transform-origin: 144.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) comprises all of the prime numbers up to 17. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.158px 7.91667px; transform-origin: 383.158px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHe told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 365.35px 7.91667px; transform-origin: 365.35px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 78.9583px 7.91667px; transform-origin: 78.9583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e*The story continues in a \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.youtube.com/watch?v=aCq04N9it8U\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eNumberphile video\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 232.967px 7.91667px; transform-origin: 232.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = RuthAaron(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 5;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 25;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 51;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 78;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 101;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 153;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 222;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 493;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 713;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 714;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 715;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 716;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 24024;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 154843;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 977779;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 6905998;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 29817473;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 89460293;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 89460294;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 89460295;\r\nassert(isequal(RuthAaron(n),'RA'))\r\n\r\n%%\r\nn = 89460296;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%  \r\nn = 89460297;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 144426976;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 333333333;\r\nassert(isequal(RuthAaron(n),'X'))\r\n\r\n%%\r\nn = 715159952;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 888681984;\r\nassert(isequal(RuthAaron(n),'A'))\r\n\r\n%%\r\nn = 1925922590;\r\nassert(isequal(RuthAaron(n),'R'))\r\n\r\n%%\r\nn = 1925922591;\r\nassert(isequal(RuthAaron(n),'A'))","published":true,"deleted":false,"likes_count":3,"comments_count":3,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":29,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-11T02:03:10.000Z","updated_at":"2026-05-25T05:46:26.000Z","published_at":"2021-06-11T02:14:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAfter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/watch?v=QjqYThEVoSQ\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHank Aaron hit his 715\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eth\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e home run\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and passed Babe Ruth’s total of 714, Carl Pomerance noticed that the union of the prime factors of the numbers (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"714 = 2*3*7*17\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e714 = 2\\\\cdot 3\\\\cdot 7 \\\\cdot 17\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"715 = 5*11*13\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e715 = 5\\\\cdot 11 \\\\cdot 13\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) comprises all of the prime numbers up to 17. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHe told a colleague that he found an interesting property of the two numbers, and when the colleague posed the problem to a class, a student observed that the sums of the prime factors of these two consecutive numbers were equal—in this case, 29. Prof. Pomerance and colleagues named numbers with this property “Ruth-Aaron numbers.”*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to classify numbers in this way. In computing sums, include only the unique factors. If a number is the lesser of a Ruth-Aaron pair, return ‘R’. If it is the greater number, return ‘A’. If multiple pairs overlap, return ‘RA’ for numbers that can serve as either the lesser or greater. If the number is not part of a Ruth-Aaron pair, return ‘X’.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e*The story continues in a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/watch?v=aCq04N9it8U\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNumberphile video\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e on this topic, and I highly recommend that you watch it. Go for the number theory and stay for the life lessons!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":61322,"title":"Determine if a number is prime","description":"Given a positive integer n, return 1 if it is prime, 0 otherwise. Example: is_prime(7) returns 1, is_prime(4) returns 0","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 42.4px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 244px 21.2px; transform-origin: 244px 21.2px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 220px 21.2px; text-align: left; transform-origin: 220px 21.2px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a positive integer n, return 1 if it is prime, 0 otherwise. Example: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eis_prime(7)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eis_prime(4)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function result = is_prime(n)\r\n  result = 0;\r\nend","test_suite":"%% Test 1\r\nassert(isequal(is_prime(7), 1))\r\n%% Test 2\r\nassert(isequal(is_prime(4), 0))\r\n%% Test 3\r\nassert(isequal(is_prime(2), 1))\r\n%% Test 4\r\nassert(isequal(is_prime(1), 0))\r\n%% Test 5\r\nassert(isequal(is_prime(13), 1))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":5047536,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":24,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2026-05-04T22:55:10.000Z","updated_at":"2026-07-20T09:19:27.000Z","published_at":"2026-05-04T22:55:10.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a positive integer n, return 1 if it is prime, 0 otherwise. Example: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eis_prime(7)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eis_prime(4)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44229,"title":"How brilliant are you?","description":"A Brilliant number is defined as a number with two prime factors, both of which have the same number of digits.  Some examples:\r\n\r\n10=2*5.  Since 2 and 5 have the same number of digits, 10 is a brilliant number.\r\n\r\n22=2*11.  Although there are two prime factors, they have a different number of digits, so 22 is not a brilliant number.\r\n\r\n30=2*3*5.  Although each prime factor has the same number of digits, there are more than two of them, so 30 is not a brilliant number.\r\n\r\nGiven a number, write a MATLAB script to determine if the number is brilliant or not.","description_html":"\u003cp\u003eA Brilliant number is defined as a number with two prime factors, both of which have the same number of digits.  Some examples:\u003c/p\u003e\u003cp\u003e10=2*5.  Since 2 and 5 have the same number of digits, 10 is a brilliant number.\u003c/p\u003e\u003cp\u003e22=2*11.  Although there are two prime factors, they have a different number of digits, so 22 is not a brilliant number.\u003c/p\u003e\u003cp\u003e30=2*3*5.  Although each prime factor has the same number of digits, there are more than two of them, so 30 is not a brilliant number.\u003c/p\u003e\u003cp\u003eGiven a number, write a MATLAB script to determine if the number is brilliant or not.\u003c/p\u003e","function_template":"function y = brilliant(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nassert(isequal(brilliant(4),1))\r\n%%\r\nassert(isequal(brilliant(8),0))\r\n%%\r\nassert(isequal(brilliant(40),0))\r\n%%\r\nassert(isequal(brilliant(343),0))\r\n%%\r\nassert(isequal(brilliant(1536),0))\r\n%%\r\nassert(isequal(brilliant(1537),1))\r\n%%\r\nassert(isequal(brilliant(49165),0))\r\n%%\r\nassert(isequal(brilliant(657721),1))\r\n%%\r\nassert(isequal(brilliant(768819),0))\r\n%%\r\nassert(isequal(brilliant(13717421),1))\r\n%%\r\nassert(isequal(brilliant(123456789),0))\r\n%%\r\nassert(isequal(brilliant(669562601),1))\r\n%%\r\nassert(isequal(brilliant(1234567890),0))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":66,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2017-06-01T18:34:04.000Z","updated_at":"2026-02-24T14:03:59.000Z","published_at":"2017-06-01T18:34:04.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA Brilliant number is defined as a number with two prime factors, both of which have the same number of digits. Some examples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e10=2*5. Since 2 and 5 have the same number of digits, 10 is a brilliant number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e22=2*11. Although there are two prime factors, they have a different number of digits, so 22 is not a brilliant number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e30=2*3*5. Although each prime factor has the same number of digits, there are more than two of them, so 30 is not a brilliant number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a number, write a MATLAB script to determine if the number is brilliant or not.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":47798,"title":"Check if the input is a balanced prime","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 90.1425px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.491px 45.0712px; transform-origin: 406.497px 45.0712px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20.5882px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10.2941px; text-align: left; transform-origin: 383.496px 10.2941px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMore about Balanced Primes on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Balanced_prime\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eWikipedia\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 59.5588px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 403.493px 29.7794px; transform-origin: 403.498px 29.7794px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisBalancedPrime(3) = false;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisBalancedPrime(7) = false;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 19.8529px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.99954px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.99954px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.99954px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.99954px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.493px 9.92647px; transform-origin: 403.498px 9.92647px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisBalancedPrime(5) = true; \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration: none; text-decoration-color: rgb(2, 128, 9); \"\u003e% since  (3+7)/2 = 5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = isBalancedPrime(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 2;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n%%\r\nx = 3;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n%%\r\nx = 5;\r\ny_correct = 1;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n%%\r\nx = 9;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nlist = [5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103];\r\nfor idx = 1:50\r\n    x = randi([10 1103]);\r\n        if ismember(x,list)\r\n            y_correct = 1;\r\n        else\r\n            y_correct = 0;\r\n        end\r\n    assert(isequal(isBalancedPrime(x),y_correct))\r\nend\r\n\r\n\r\n%%\r\nx = 81001;\r\ny_correct = 1;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 81402;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 177487;\r\ny_correct = 1;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 467101;\r\ny_correct = 1;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 467102;\r\ny_correct = 0;\r\nassert(isequal(isBalancedPrime(x),y_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":8703,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":30,"test_suite_updated_at":"2020-12-04T16:29:03.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-04T16:26:34.000Z","updated_at":"2026-05-30T19:13:44.000Z","published_at":"2020-12-04T16:26:34.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMore about Balanced Primes on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Balanced_prime\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eWikipedia\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[isBalancedPrime(3) = false;\\nisBalancedPrime(7) = false;\\nisBalancedPrime(5) = true; % since  (3+7)/2 = 5]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44748,"title":"Amicable numbers","description":"Test whether two numbers are \u003chttps://en.wikipedia.org/wiki/Amicable_numbers amicable\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\r\n\r\n\r\nExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.","description_html":"\u003cp\u003eTest whether two numbers are \u003ca href = \"https://en.wikipedia.org/wiki/Amicable_numbers\"\u003eamicable\u003c/a\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\u003c/p\u003e\u003cp\u003eExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.\u003c/p\u003e","function_template":"function y = amicable(m,n)\r\n  y = false;\r\nend","test_suite":"%%\r\nm = 220; n = 284;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 220; n = 504;\r\ny_correct = false;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 2620; n = 2924;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 5020; n = 5564;\r\ny_correct = true;\r\nassert(isequal(amicable(m,n),y_correct))\r\n\r\n%%\r\nm = 2924; n = 5020;\r\ny_correct = false;\r\nassert(isequal(amicable(m,n),y_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":4,"created_by":254267,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":80,"test_suite_updated_at":"2018-10-22T18:10:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-10-22T17:57:52.000Z","updated_at":"2026-06-08T05:35:58.000Z","published_at":"2018-10-22T18:02:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTest whether two numbers are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Amicable_numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eamicable\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, meaning that the sum of the proper divisors of each number is equal to the other number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample: 220 and 284 are amicable numbers because the proper divisors of 220 are 1,2,4,5,10,11,20,22,44,55,110 and their sum is 284, while the proper divisors of 284 are 1,2,4,71,142 and their sum is 220.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60972,"title":"Check Bertrand-Chebyshev's theorem about prime numbers","description":"Historical context\r\nBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than 1, there always exists at least on prime number between it and its double : \r\n\r\n\r\n\r\nProblem statement\r\nFor a given input n, List the prime number(s) p in the interval ]n; 2n[.\r\n\r\nExamples\r\n\r\nn = 2 =\u003e p = 3;\r\nn = 3 =\u003e p = 5;\r\nn = 7 =\u003e  p = [11, 13];\r\nn = 47 =\u003e p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 697.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 348.733px; transform-origin: 408px 348.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 385px 8px; transform-origin: 385px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 223.658px 8px; transform-origin: 223.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethere always exists at least on prime number between it and its double : \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; 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4ZzqN0yWHGoepV+Us28EyJid++zV7veNaqJdEN+l6q5dK6srhjjXxaf1QoK5xrH6vdEEQr3dZOQRgF2DGsoFAoHAeASC+MZjGiUGAoFAIBAI7BiBIL4dd06oFggEAoFAIDAegSC+8ZhGiYFAIBAIBAI7RuA/zuJmZAlv4pMAAAAASUVORK5CYII=\" width=\"223\" height=\"19.5\" style=\"width: 223px; height: 19.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.0667px 8px; transform-origin: 54.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given input \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.675px 8px; transform-origin: 81.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, List the prime number(s) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.8917px 8px; transform-origin: 66.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the interval ]n; 2n[.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.6833px 8px; transform-origin: 46.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 2 =\u0026gt; p = 3;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.6833px 8px; transform-origin: 46.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 3 =\u0026gt; p = 5;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 68.3333px 8px; transform-origin: 68.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 7 =\u0026gt;  p = [11, 13];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 152.467px 8px; transform-origin: 152.467px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 47 =\u0026gt; p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = check_Bertrand_Chebyshev_theorem(n)\r\n    p = n;\r\nend","test_suite":"%%\r\nn = 2;\r\np_correct = 3;\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 3;\r\np_correct = 5;\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 7;\r\np_correct = [11, 13];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 11;\r\np_correct = [13, 17, 19];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 47;\r\np_correct = [53, 59, 61, 67, 71, 73, 79, 83, 89];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_Bertrand_Chebyshev_theorem.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:03:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":41,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:28:51.000Z","updated_at":"2026-07-20T06:40:55.000Z","published_at":"2025-07-21T04:30:08.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethere always exists at least on prime number between it and its double : \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall n \\\\in \\\\mathbb{N}^* - \\\\{ 1 \\\\}, \\\\exists p \\\\in \\\\mathbb{P}, \\\\mathbf{n \u0026lt; p \u0026lt; 2n}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given input \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, List the prime number(s) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in the interval ]n; 2n[.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 2 =\u0026gt; p = 3;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 3 =\u0026gt; p = 5;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 7 =\u0026gt;  p = [11, 13];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 47 =\u0026gt; p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60946,"title":"Fermat primes","description":"Historical context\r\n\r\n\r\nThe french mathematician Pierre de Fermat found the formula  to find some prime numbers. He thought / conjectured all these numbers where primes, but P(5) is actually the first not prime Fermat number.\r\n\r\n\r\nProblem statement\r\n\r\nFor a given integer m, , list the Fermat prime numbers less or equal to m in a vector fp. Also compute their frequency / ratio f compare to all the primes in the same range. If  , your function should return fp = [] and f = 0.\r\n\r\nExamples\r\n\r\nIf m = 10 then fp = [3, 5] and f = 2/3 since 3 = P(0), 5 = P(1), and there are in total three primes in the range [3; 10] ;\r\nIf m = 300 then fp = [3, 5, 17, 257] and f = 4/61 since 3 = P(0), 5 = P(1), 17 = P(2), 257 = P(3) and there are in total 61 prime numbers in the range [3; 300] ; \r\nIf m = 2 then fp = [] and f = 0;\r\n\r\n\r\nTip \r\n\r\n here.\r\n\r\n\r\nFobidden functions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\n\r\n\r\nSee also\r\nProblem 60942. Euler primes\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1001.03px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 500.517px; transform-origin: 408px 500.517px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 22.5px; text-align: left; transform-origin: 385px 22.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 193.7px 8px; transform-origin: 193.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe french mathematician Pierre de Fermat found the formula \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMAAAAAvCAYAAAC1zsVQAAAHSUlEQVR4Xu1cWehvUxS+N6IMmR4MiQcezEVcZIiEMjx4MDxIN+oawu12Izzo5oEML4ZMebhKmZKUOVEkY6KIBx7IlDJliBDfp71r/Vd7n7PPPvvM69TX//c7v7PX3nvt9e219tr7/Fevsss0sGANrF5w363rpoFVRgAzgkVrwAiw6OG3zhsBzAYWrQEjwKKH3zpvBDAbWLQGjACLHn7rvBHAbGDRGjACLHr4/+/8C8ApQg2n4vNGd+9X/L0RuGmuajICzHVkm/XrX/H4J/h8O3A9sLu7fx7+PtpM5DSeNgJMY5y6bqUkwL6o7DPgFuAqV/H9+Htx140YQr4RYAitj69OSYBd0LwfjADjGyRrUZ4GLlHhzGP4fqkzci/RCJCn21albkDpA4CbgXdaSVpZmAu684Fn5hq3NtCVDGNksW/wZQ9xwwjQQKklHn0SQo4F3gdOLiFQyNgZn18B9nZ/zyosfyri9kFDPwUYv78MbACOFI2nF7jXfTcCOEVod1k12EyRfQR8ADwBvJhoGTT+k1zZoxPLNH2MJGDbdgSeB+ZCAo4P+3IwsL3r41P4ex/AuF1ezNpwgpEpzK/x3Wd25MJ2jgQ4F30l6V8Fro4ZUGwRfC0KMP+rrzW4wXCFs8vdgMwfv4XvV7jfY/XRJV8ObAnsFhi0WLmc+yTBF8DfwDWAn+1yZA1dhn15FpAzuGwTQ5rjAGZv/MWxeVfpWIZE1+E3koOyvxflprwIZl+YraIdeqLfis+NCaCVQv3QwI9SlvCmGhR6BYYeejZiMQ4IN10421wG9GGQXGdwoP8CDlIGoroy6q9az6HGhsZHP0dPsc7d9JOZlk053AOgV99OCOAGWaqX71uZnJA5YZ+m2sx2ZBGABaVb5PeQgkOeglkGuh990XMcBvwCMDTp6/oJFTFceA84oq9KC9bDsOcep3+6c15nAvsF6qgz0o9dudgYFWx2b6JogycADK1PBM5RNXdKAD+ryzp1hoG/DTH7+zbdgQ90ibz8rNfb6BSoiHE7PZn2mIzxmwy2HwPu9J4ByHCpQDOTRPhJNcVbJQkMPKQn7t4JwDbptQVDH2Z7eN/HmLkdbFrOZ0OomMeBkHdqKrOv59nWCwHO7KFLLmr5e2ywGdJ+CPw8oPGzfYsgANcBDDn8JdcTX+Hmnn1Zj6iHYdAOwO/AtgPUn1slyfsjEFpTUaaM6asIQG9xiDL+Otm5ba4qtwgCcKEkZyzOYo84rTyIv2sjGmJ8zpSeX737x3xcywyGj31JMmZF9I5mTPlv4Ae/gJ9iGBTrl97oCh1aY4x8gTJ+yuN6gPdLbkLWkWZ2BAjFoXohJgepMh6D9ugtPgdk9oGDugkILfoYz+5fp3X8LtsgN4ASio76Eal/7XnZcL+App4Y/vhrL3ygl45l7Lrq9KwI4JUrlRUyLnnevC5LQVmhtNxm3GfmYifgNUB6iRSDbkLC0ODHjhTkGErJA4hyDeDz+r5N9KhvVzQwdfLI6WOszKQJwOzOQ27W4KaLNEIqcz0QyhNLg84hgPYaOu7VIVdI+TJblZMCHCMBZGipsyqM77kLLD2p1ssQR5xnQQCpSC7QXgKq4sjSBNDGmJJSkwRIeV4bC73d2kJTod5MzBXrc/qcfI4BYgvlXPldlJs0AXIMh0qcAwG6MIY2Mv0G5NiMX+fgc/uYa2usr9d9gJQOjoEAMlwYwvWn6Cn1Gcb2PN355QhnfiNAYBRlpqLEGiAnBGq7CE41zq6fY5bsdVdJLOxhuNbHOatQXznZVV3+MJ8/SRx7lieMc1/BHJ0HaGp8OgukF8FtCZDzgvdYFsF+Qytm/FzrXASMdbd7kWsAuQDlW1o8h1J1dUEAv2Bkvf6l7yaz9RgIwOzX8ZGwh56BR00eAMb8b0wWSQAa2m/ANkDKUYguCPAn6t4K+A7YtYnlu2eHzgKF9lxi3RjzTvdkCOAPkEklh3YaU22JrvtsgJtAdTNwaQJID1S3E53anz6fa2L8bFfJjbbS/RyCAJVJj5iy9GaTV0Tu7CKN8E4IuzKi2dBRCN0BffyibkfzadR1uquvjnylB7ytPJ/xqdrQknW0SR+2bWtK+a4JEHo/hRu4fBkquE+iCZDyTjAFhs6n1ymg7oWY2GE4yvUkCJ094u8kQehgFwn1LUDFbwZyMwt1fevqd748lGr8Uk9dtaet3C4I4MNTnm/SByl9e+X76ytsoE93ybCKZ9K3Bu4CYl6grZJl+YfxhRmRPwAew57CbmnJ/o9NVhcEaNXHPgnAhpKttwFbAMxodHkUl2HXc8A/ADNPY32ftdUATqyw3ydok+cv2uW+CcDGM6VIN8SXUw7saFae03+EKDrgJmylBoYggPcEmxwJDi9MAho//x0I066sY6hdUbO1CWhgKAJQNQxRDgV4rqVkKNSV3AkMpzWxqQaGJEDTttrzpoHiGjACFFepCZySBowAUxota2txDRgBiqvUBE5JA0aAKY2WtbW4Bv4D2czmP1crQekAAAAASUVORK5CYII=\" width=\"96\" height=\"23.5\" style=\"width: 96px; height: 23.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.625px 8px; transform-origin: 132.625px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to find some prime numbers. He thought / conjectured all these numbers where primes, but P(5) is actually the first not prime Fermat number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 144.3px 8px; transform-origin: 144.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Fermat prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.1083px 8px; transform-origin: 38.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 98.6167px 8px; transform-origin: 98.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 146.242px 8px; transform-origin: 146.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.8583px 8px; transform-origin: 82.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23.1333px 8px; transform-origin: 23.1333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e fp = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 10\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.5833px 8px; transform-origin: 28.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 2/3 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 56.0083px 8px; transform-origin: 56.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e3 = P(0), 5 = P(1),\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 169.192px 8px; transform-origin: 169.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and there are in total three primes in the range [3; 10] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2.33333px 8px; transform-origin: 2.33333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 300\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 55.8083px 8px; transform-origin: 55.8083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 17, 257]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 4/61 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 86.9333px 8px; transform-origin: 86.9333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e3 = P(0), 5 = P(1), 17 = P(2)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.875px 8px; transform-origin: 32.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e257 = P(3)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 66.125px 8px; transform-origin: 66.125px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand there are in total 61\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 31.1px 8px; transform-origin: 31.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [3; 300] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.0333px 8px; transform-origin: 20.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.25px 8px; transform-origin: 19.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003efp = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60942-euler-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60942. Euler primes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [fp, f] = fermat_primes(n)\r\n  fp = n;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 10;\r\nfp_correct = [3, 5];\r\nf_correct = 2/3;\r\n[fp,f] = fermat_primes(m);\r\nassert(isequal([fp,f],[fp_correct,f_correct]));\r\n\r\n%%\r\nm = 300;\r\nfp_correct = [3, 5, 17, 257];\r\nf_correct = 4/61;\r\n[fp,f] = fermat_primes(m);\r\nassert(isequal([fp,f],[fp_correct,f_correct]));\r\n\r\n%%\r\nm = 2;\r\nfp_correct = [];\r\nf_correct = 0;\r\n[fp,f] = fermat_primes(m);\r\nassert(isequal([fp,f],[fp_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('fermat_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:52:07.000Z","deleted_by":null,"deleted_at":null,"solvers_count":21,"test_suite_updated_at":"2025-07-09T05:57:08.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-27T05:33:53.000Z","updated_at":"2026-06-09T14:35:13.000Z","published_at":"2025-06-27T06:14:34.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe french mathematician Pierre de Fermat found the formula \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{P(n) = 2^{2^n} + 1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to find some prime numbers. He thought / conjectured all these numbers where primes, but P(5) is actually the first not prime Fermat number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em \\\\geq 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Fermat prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em \u0026lt; 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e fp = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 2/3 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3 = P(0), 5 = P(1),\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and there are in total three primes in the range [3; 10] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 300\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 17, 257]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 4/61 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3 = P(0), 5 = P(1), 17 = P(2)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e257 = P(3)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand there are in total 61\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [3; 300] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efp = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en \\\\leq 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden 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w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60942-euler-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60942. Euler primes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":47808,"title":"Check if the input is an interprime","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 70.8px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 331.5px 35.4px; transform-origin: 331.5px 35.4px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 308.5px 10.4px; text-align: left; transform-origin: 308.5px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMore information about interprimes on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Interprime\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eWikipedia\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 40px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 328.5px 20px; transform-origin: 328.5px 20px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.5px 10px; transform-origin: 328.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisInterprime(2) = 0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.5px 10px; transform-origin: 328.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eisInterprime(4) = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = isInterprime(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 2;\r\ny_correct = 0;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n%%\r\ninterprimeList = [4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99];\r\nfor idx = 1:100\r\n    x = randi([2 99]);\r\n    y_correct = 0;\r\n    if ismember(x,interprimeList)\r\n        y_correct = 1;\r\n    end\r\n    assert(isequal(isInterprime(x),y_correct))\r\nend\r\n\r\n%%\r\nx = 4676;\r\ny_correct = 1;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 46725;\r\ny_correct = 1;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 4622;\r\ny_correct = 0;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n%%\r\nx = 467475;\r\ny_correct = 1;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n\r\n%%\r\nx = 467476;\r\ny_correct = 0;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n%% \r\nx = 499994;\r\ny_correct = 1;\r\nassert(isequal(isInterprime(x),y_correct))\r\n\r\n    ","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":8703,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":"2020-12-28T11:41:15.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2020-12-04T17:23:08.000Z","updated_at":"2026-05-30T19:13:49.000Z","published_at":"2020-12-04T17:23:08.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMore information about interprimes on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Interprime\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eWikipedia\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[isInterprime(2) = 0;\\nisInterprime(4) = 1;]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":56743,"title":"Count the unitary divisors of a number","description":"Cody Problem 56738 asks for a list of the unitary divisors of a number. For this problem, write a function to count the unitary divisors of a number. For example, the unitary divisors of 18 are 1, 2, 9, and 18; therefore, the number of unitary divisors is 4.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 63px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 31.5px; transform-origin: 407px 31.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/56738-list-unitary-divisors-of-a-number\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 56738\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 318.25px 8px; transform-origin: 318.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks for a list of the unitary divisors of a number. For this problem, write a function to count the unitary divisors of a number. For example, the unitary divisors of 18 are 1, 2, 9, and 18; therefore, the number of unitary divisors is 4.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = countUDivisors(n)\r\n  y = length(n(y/n==true));\r\nend","test_suite":"%%\r\nn = 18;\r\ny_correct = 4;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 128;\r\ny_correct = 2;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 996;\r\ny_correct = 8;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 1228;\r\ny_correct = 4;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 54321;\r\ny_correct = 8;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 648207;\r\ny_correct = 8;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 840519372;\r\ny_correct = 128;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = 7420738134810;\r\ny_correct = 4096;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = flintmax/2-7;\r\ny_correct = 8;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%%\r\nn = flintmax-4;\r\ny_correct = 64;\r\nassert(isequal(countUDivisors(n),y_correct))\r\n\r\n%% \r\np = primes(randi(1000000));\r\nn = p(end);\r\nassert(isequal(countUDivisors(n),2))\r\n\r\n%%\r\nfiletext = fileread('countUDivisors.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-11-24T17:08:57.000Z","deleted_by":null,"deleted_at":null,"solvers_count":15,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-11-24T17:07:22.000Z","updated_at":"2026-07-09T14:40:21.000Z","published_at":"2022-11-24T17:08:57.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/56738-list-unitary-divisors-of-a-number\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 56738\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks for a list of the unitary divisors of a number. For this problem, write a function to count the unitary divisors of a number. For example, the unitary divisors of 18 are 1, 2, 9, and 18; therefore, the number of unitary divisors is 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":2733,"title":"Evil Number","description":"Check if a given natural number is evil or not. \r\n\r\nRead more at \u003chttps://oeis.org/A001969 OEIS\u003e.","description_html":"\u003cp\u003eCheck if a given natural number is evil or not.\u003c/p\u003e\u003cp\u003eRead more at \u003ca href = \"https://oeis.org/A001969\"\u003eOEIS\u003c/a\u003e.\u003c/p\u003e","function_template":"function tf = isevil(n)\r\n  tf = ;\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = false;\r\nassert(isequal(isevil(x),y_correct))\r\n%%\r\nx = 3;\r\ny_correct = true;\r\nassert(isequal(isevil(x),y_correct))\r\n%%\r\nx = [18, 20, 23, 24, 27, 45, 46, 48, 96, 99, 123,];\r\ny_correct = true;\r\nassert(isequal(all(arrayfun(@isevil,x)),y_correct))\r\n%%\r\nx = [14, 16, 19, 37, 38, 55, 56, 59, 62,  79, 82, 91, 93, 94, 97, 98, 117, 118, 121];\r\ny_correct = false;\r\nassert(isequal(any(arrayfun(@isevil,x)),y_correct))\r\n%%\r\nx = 2^randi([5 10])+1;\r\ny_correct = true;\r\nassert(isequal(isevil(x),y_correct))\r\n\r\n%%\r\n% more test cases may be introduced\r\n%%\r\n% DISABLED\r\n% ________'FAIR'_SCORING_SYSTEM______________\r\n%\r\n% This section scores for usage of ans\r\n% and strings, which are common methods \r\n% to reduce cody size of solution.\r\n% Here, strings are threated like vectors.\r\n% Please do not hack it, as this problem\r\n% is not mentioned to be a hacking problem.\r\n% \r\n  try\r\n% disable:\r\nassert(false) \r\n%\r\n  size_old = feval(@evalin,'caller','score');\r\n%\r\n  all_nodes = mtree('isevil.m','-file');\r\n  str_nodes = mtfind(all_nodes,'Kind','STRING');\r\n   eq_nodes = mtfind(all_nodes,'Kind','EQUALS');\r\nprint_nodes = mtfind(all_nodes,'Kind','PRINT');\r\n expr_nodes = mtfind(all_nodes,'Kind','EXPR');\r\n%\r\n       size = count(all_nodes)           ...\r\n              +sum(str_nodes.nodesize-1) ...\r\n              +2*(count(expr_nodes)      ...\r\n                  +count(print_nodes)    ...\r\n                  -count(eq_nodes));\r\n%\r\n  feval(@assignin,'caller','score',size);\r\n%\r\n  fprintf('Size in standard cody scoring is %i.\\n',size_old);\r\n  fprintf('Here it is %i.\\n',size);\r\n  end\r\n%\r\n%_________RESULT_____________________________","published":true,"deleted":false,"likes_count":3,"comments_count":6,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":276,"test_suite_updated_at":"2016-12-26T10:21:47.000Z","rescore_all_solutions":true,"group_id":8,"created_at":"2014-12-07T21:50:01.000Z","updated_at":"2026-06-08T04:55:31.000Z","published_at":"2015-01-19T12:47:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck if a given natural number is evil or not.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRead more at\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://oeis.org/A001969\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOEIS\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60942,"title":"Euler primes","description":"Historical context\r\n\r\n\r\nThe great swiss mathematician Leonhard Euler found the formula  to give some prime numbers.\r\n\r\n\r\nProblem statement\r\n\r\nFor a given integer m, m \u003e= 41, list the Euler prime numbers less or equal to m in a vector ep. Also compute their frequency / ratio f compare to all the primes in the same range. If m is not a positive integer greater or equal to 41, your function should return ep = [] and f = 0.\r\n\r\nExamples (check the tests for more)\r\n\r\nIf m = 41 then ep = 41 and f = 1 since 41 = P(0) and it is the only prime number m greater or equal and less or equal to 41;\r\nIf m = 100 then ep = [41, 43, 47, 53, 61, 71, 83, 97] and f = 8/13 since 41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), and in total there are 13 prime numbers in the range [41; 100] ; \r\nIf m = 19 then ep = [] and f = 0;\r\n\r\n\r\nTip \r\n\r\nn is less or equal to 107 here.\r\n\r\n\r\nFobidden functions\r\n\r\nregexp\r\nstr2num\r\nassignin\r\n\r\nSee also\r\nProblem 60944. Ulam primes first formula, Problem 60943. Ulam primes second formula\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 988.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 494.233px; transform-origin: 408px 494.233px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 204.608px 8px; transform-origin: 204.608px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe great swiss mathematician Leonhard Euler found the formula \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"125\" height=\"19.5\" style=\"width: 125px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.1833px 8px; transform-origin: 92.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to give some prime numbers.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.075px 8px; transform-origin: 26.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u0026gt;= 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 138.475px 8px; transform-origin: 138.475px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Euler prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.1667px 8px; transform-origin: 36.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 8px; transform-origin: 7.78333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.3833px 8px; transform-origin: 96.3833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 286.658px 8px; transform-origin: 286.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.6667px 8px; transform-origin: 11.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e41, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.35px 8px; transform-origin: 63.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.6917px 8px; transform-origin: 24.6917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e ep = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3333px 8px; transform-origin: 79.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests for more)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 51.0833px; transform-origin: 392px 51.0833px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = 41\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.1417px 8px; transform-origin: 16.1417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.9833px 8px; transform-origin: 28.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e41 = P(0)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 99.575px 8px; transform-origin: 99.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and it is the only prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.533px 8px; transform-origin: 108.533px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e greater or equal and less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10.1167px 8px; transform-origin: 10.1167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e41;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 100\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 110.258px 8px; transform-origin: 110.258px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = [41, 43, 47, 53, 61, 71, 83, 97]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 8/13 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 141.192px 8px; transform-origin: 141.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 64.1833px 8px; transform-origin: 64.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand in total there are\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.6667px 8px; transform-origin: 11.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e 13 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.9917px 8px; transform-origin: 34.9917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [41; 100] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 19 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.8083px 8px; transform-origin: 20.8083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.3417px 8px; transform-origin: 58.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e107\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60944-ulam-primes-first-formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eProblem 60944. Ulam primes first formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60943-ulam-primes-second-formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eProblem 60943. Ulam primes second formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ep,f] = euler_primes(m)\r\n  ep = m;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 41;\r\nep_correct = 41;\r\nf_correct = 1;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 100;\r\nep_correct = [41, 43, 47, 53, 61, 71, 83, 97];\r\nf_correct = 8/13;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 200;\r\nep_correct = [41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197];\r\nf_correct = 13/34;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 19;\r\nep_correct = [];\r\nf_correct = 0;\r\n[ep,f] = euler_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('euler_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:49:26.000Z","deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":"2025-07-09T05:56:34.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-26T14:44:30.000Z","updated_at":"2026-05-16T04:50:50.000Z","published_at":"2025-06-26T15:44:04.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe great swiss mathematician Leonhard Euler found the formula \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{P(n) = n^2 + n + 41}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to give some prime numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u0026gt;= 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Euler prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e ep = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = 41\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41 = P(0)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and it is the only prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e greater or equal and less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 100\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = [41, 43, 47, 53, 61, 71, 83, 97]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 8/13 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e41 = P(0), 43 = P(1), 47 = P(2), 53 = P(3), 61 = P(4), 71 = P(5), 83 = P(6), and 97 = P(7), \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand in total there are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e 13 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [41; 100] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 19 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e107\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e here.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60944-ulam-primes-first-formula\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem 60944. Ulam primes first formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60943-ulam-primes-second-formula\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem 60943. Ulam primes second formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60997,"title":"Multiply real numbers to get the smallest integers","description":"Write a function that takes a vector of real numbers and multiplies them by a factor to produce the integers with the smallest absolute value. For example, if the input vector is [3/4 5/6 7/10], the output should be [45 50 42]; that is, the input vector should be multiplied by 60.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 63px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 31.5px; transform-origin: 408px 31.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes a vector of real numbers and multiplies them by a factor to produce the integers with the smallest absolute value. For example, if the input vector is [3/4 5/6 7/10], the output should be [45 50 42]; that is, the input vector should be multiplied by 60.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = integerize(x)\r\n  y = ceil(x);\r\nend","test_suite":"%%\r\nx = [3/4 5/6 7/10];\r\ny = integerize(x);\r\ny_correct = [45 50 42];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nx = [1.5 8.9 6.35 7.125];\r\ny = integerize(x);\r\ny_correct = [60 356 254 285];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nx = randi(80,[1 14])-40;\r\ny = integerize(x);\r\ny_correct = x;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nx = [4/5 -8/7 15/12 -13/49 83/56];\r\ny = integerize(x);\r\ny_correct = [1568 -2240 2450 -520 2905];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm = randi(6);\r\np = primes(100);\r\nk = randperm(length(p),2*m);\r\nn = p(k(1:m));\r\nd = p(k(m+1:end));\r\nf = prod(d);\r\nx = n./d;\r\ny = integerize(x);\r\ny_correct = x*f;\r\nassert(isequal(y,y_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2025-09-08T01:51:47.000Z","deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-09-08T01:51:22.000Z","updated_at":"2026-06-06T06:02:53.000Z","published_at":"2025-09-08T01:51:47.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes a vector of real numbers and multiplies them by a factor to produce the integers with the smallest absolute value. For example, if the input vector is [3/4 5/6 7/10], the output should be [45 50 42]; that is, the input vector should be multiplied by 60.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60975,"title":"Check Legendre's conjecture","description":"Context and problem statement\r\nLegendre's conjecture states for a given positive integer n, there always exists at least one prime number p between n² and (n+1²) :\r\n\r\n\r\n\r\nFor a given input n, List the prime number(s) p in the interval ]n^2; (n+1)^2[.\r\n\r\nExamples\r\n\r\nn = 1   =\u003e p = [2, 3];\r\nn = 2   =\u003e p = [5, 7];\r\nn = 10 =\u003e p = [101, 103, 107, 109, 113];\r\nn = 20 =\u003e p = [401, 409, 419, 421, 431, 433, 439];\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nProblem 60972. Check Bertrand-Chebyshev's theorem about prime numbers\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 697.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 348.733px; transform-origin: 408px 348.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 105px 8px; transform-origin: 105px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eContext and problem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.825px 8px; transform-origin: 174.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eLegendre's conjecture states for a given positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 144.308px 8px; transform-origin: 144.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethere always exists at least one prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3417px 8px; transform-origin: 30.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and (n+1²) :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"224\" height=\"19.5\" style=\"width: 224px; height: 19.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.0667px 8px; transform-origin: 54.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given input \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.675px 8px; transform-origin: 81.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, List the prime number(s) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 90px 8px; transform-origin: 90px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the interval ]n^2; (n+1)^2[.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 1   =\u0026gt; p = [2, 3];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 2   =\u0026gt; p = [5, 7];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 124.742px 8px; transform-origin: 124.742px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 10 =\u0026gt; p = [101, 103, 107, 109, 113];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 156.375px 8px; transform-origin: 156.375px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 20 =\u0026gt; p = [401, 409, 419, 421, 431, 433, 439];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60972\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60972. Check Bertrand-Chebyshev's theorem about prime numbers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = check_legendre_conjecture(n)\r\n  p = n;\r\nend","test_suite":"%%\r\nn = 1\r\np_correct = [2, 3];\r\nassert(isequal(check_legendre_conjecture(n),p_correct))\r\n\r\n%%\r\nn = 2\r\np_correct = [5, 7];\r\nassert(isequal(check_legendre_conjecture(n),p_correct))\r\n\r\n%%\r\nn = 10\r\np_correct = [101, 103, 107, 109, 113];\r\nassert(isequal(check_legendre_conjecture(n),p_correct))\r\n\r\n%%\r\nn = 20\r\np_correct = [401, 409, 419, 421, 431, 433, 439];\r\nassert(isequal(check_legendre_conjecture(n),p_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_legendre_conjecture.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:05:15.000Z","deleted_by":null,"deleted_at":null,"solvers_count":43,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-21T08:10:57.000Z","updated_at":"2026-07-20T06:46:32.000Z","published_at":"2025-07-21T08:28:05.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eContext and problem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLegendre's conjecture states for a given positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethere always exists at least one prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and (n+1²) :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall n \\\\in \\\\mathbb{N}^*, \\\\exists p \\\\in \\\\mathbb{P}, \\\\mathbf{n^2 \u0026lt; p \u0026lt; (n+1)^2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given input \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, List the prime number(s) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in the interval ]n^2; (n+1)^2[.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 1   =\u0026gt; p = [2, 3];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 2   =\u0026gt; p = [5, 7];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 10 =\u0026gt; p = [101, 103, 107, 109, 113];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 20 =\u0026gt; p = [401, 409, 419, 421, 431, 433, 439];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60972\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60972. Check Bertrand-Chebyshev's theorem about prime numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60970,"title":"Check p = 6n +/- 1, the generic formula for odd prime numbers greater than 3","description":"Problem statement\r\n\r\nFor all odd prime number p greater than 3, there exists a positive integer n, such that p = 6n +/- 1 :\r\n\r\n\r\n\r\nCheck this formula for some given odd primes in a vector by computing n for each p.\r\n\r\nExamples\r\n\r\np = 17 =\u003e n = 3;\r\n\r\np = 19 =\u003e n = 3;\r\n\r\n p = [5, 7, 11, 13, 17, 19] =\u003e n = [1, 1, 2, 2, 3, 3];\r\n\r\np = [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\r\n             =\u003e n = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12,  12, 13, 14, 15, 16];\r\n\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 919.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 459.733px; transform-origin: 408px 459.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.1333px 8px; transform-origin: 80.1333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor all odd prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.7333px 8px; transform-origin: 37.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003egreater than\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e 3\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.4583px 8px; transform-origin: 96.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, there exists a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3333px 8px; transform-origin: 30.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esuch that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep = 6n +/- 1 :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"231\" height=\"18\" style=\"width: 231px; height: 18px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 222.1px 8px; transform-origin: 222.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCheck this formula for some given odd primes in a vector by computing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.9417px 8px; transform-origin: 15.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eeach\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e p.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 17 =\u0026gt; n = 3;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 19 =\u0026gt; n = 3;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 148.817px 8px; transform-origin: 148.817px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 2, 3, 3];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 275.4px 8px; transform-origin: 275.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 266.008px 8px; transform-origin: 266.008px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e             =\u0026gt; n = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12,  12, 13, 14, 15, 16];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = check_odd_primes_gen_formula3(p)\r\n  n = p;\r\nend","test_suite":"%%\r\np = 17;\r\nn_correct = 3;\r\nassert(isequal(check_odd_primes_gen_formula3(p),n_correct))\r\n\r\n\r\n%%\r\np = 19;\r\nn_correct = 3;\r\nassert(isequal(check_odd_primes_gen_formula3(p),n_correct))\r\n\r\n\r\n%%\r\np = [5, 7, 11, 13, 17, 19];\r\nn_correct = [1, 1, 2, 2, 3, 3];\r\nassert(isequal(check_odd_primes_gen_formula3(p),n_correct))\r\n\r\n\r\n%%\r\np = [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\r\nn_correct = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12,  12, 13, 14, 15, 16];\r\nassert(isequal(check_odd_primes_gen_formula3(p),n_correct))\r\n\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_odd_primes_gen_formula3.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-20T05:22:24.000Z","deleted_by":null,"deleted_at":null,"solvers_count":42,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:25:50.000Z","updated_at":"2026-07-20T06:38:58.000Z","published_at":"2025-07-20T05:22:25.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor all odd prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003egreater than\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e 3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, there exists a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esuch that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 6n +/- 1 :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p \u0026gt; 3 \\\\Rightarrow \\\\exists n   \\\\in \\\\mathbb{N}, p = 6n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck this formula for some given odd primes in a vector by computing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eeach\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 17 =\u0026gt; n = 3;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 19 =\u0026gt; n = 3;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 2, 3, 3];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e             =\u0026gt; n = [1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12,  12, 13, 14, 15, 16];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60969,"title":"Check p = 4n +/- 1, the generic formula for odd prime numbers","description":"Problem statement\r\n\r\nFor all odd prime number p, there exists a positive integer n, such that p = 4n +/- 1 :\r\n\r\n\r\n\r\nCheck this formula for some given odd primes in a vector by computing n for each p.\r\n\r\nExamples\r\n\r\np = 17 =\u003e n = 4;\r\n\r\np = 19 =\u003e n = 5;\r\n\r\n p = [3, 5, 7, 11, 13, 17, 19] =\u003e n = [1, 1, 2, 3, 3, 4, 5];\r\n\r\np = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\r\n             =\u003e n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\r\n\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 919.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 459.733px; transform-origin: 408px 459.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.1333px 8px; transform-origin: 80.1333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor all odd prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.4583px 8px; transform-origin: 96.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, there exists a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3333px 8px; transform-origin: 30.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esuch that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep = 4n +/- 1 :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"231\" height=\"18\" style=\"width: 231px; height: 18px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 222.1px 8px; transform-origin: 222.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCheck this formula for some given odd primes in a vector by computing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.9417px 8px; transform-origin: 15.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eeach\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e p.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 17 =\u0026gt; n = 4;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 19 =\u0026gt; n = 5;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 164.367px 8px; transform-origin: 164.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 280.842px 8px; transform-origin: 280.842px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 287.408px 8px; transform-origin: 287.408px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e             =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = check_odd_primes_gen_formula(p)\r\n  n = p;\r\nend","test_suite":"%%\r\np = 17;\r\nn_correct = 4;\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = 19;\r\nn_correct = 5;\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = [3, 5, 7, 11, 13, 17, 19];\r\nn_correct = [1, 1, 2, 3, 3, 4, 5];\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\r\nn_correct = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_odd_primes_gen_formula.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-20T05:09:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":47,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:24:51.000Z","updated_at":"2026-07-20T06:37:47.000Z","published_at":"2025-07-20T05:09:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor all odd prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, there exists a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003esuch that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 4n +/- 1 :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p \u0026gt; 2 \\\\Rightarrow \\\\exists n   \\\\in \\\\mathbb{N}, p = 4n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck this formula for some given odd primes in a vector by computing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003eeach\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 17 =\u0026gt; n = 4;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 19 =\u0026gt; n = 5;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e             =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44230,"title":"I'm going to enjoy watching you calculate, Mr Anderson","description":"Smith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\r\n4+9+3+7+7+7+5=42\r\nThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\r\nSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 174px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 87px; transform-origin: 407px 87px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.5px 8px; transform-origin: 375.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSmith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e4+9+3+7+7+7+5=42\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 251px 8px; transform-origin: 251px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.5px 8px; transform-origin: 380.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = smith(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nassert(isequal(smith(4937775),1))\r\n%%\r\nassert(isequal(smith(1164),0))\r\n%%\r\nassert(isequal(smith(19683),1))\r\n%%\r\nassert(isequal(smith(11),0))  % Remember - Smith numbers are composite\r\n%%\r\nassert(isequal(smith(11^2),1))\r\n%%\r\nassert(isequal(smith(345741),1))\r\n%%\r\nassert(isequal(smith(19876),0))\r\n%%\r\nassert(isequal(smith(314159),0))\r\n%%\r\nassert(isequal(smith(612985),1))\r\n%%\r\nassert(isequal(smith(12379887),1))\r\n%%\r\nassert(isequal(smith(23456789),0))\r\n%%\r\nassert(isequal(smith(13),0))\r\n%%\r\nassert(isequal(smith(23),0))\r\n%%\r\ny=primes(randi(1e5));\r\nassert(isequal(smith(y(end)),0))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":1615,"edited_by":223089,"edited_at":"2023-01-07T08:26:27.000Z","deleted_by":null,"deleted_at":null,"solvers_count":49,"test_suite_updated_at":"2023-01-07T08:26:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2017-06-01T19:16:25.000Z","updated_at":"2026-03-16T15:29:16.000Z","published_at":"2017-06-01T19:16:25.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSmith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e4+9+3+7+7+7+5=42\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":53940,"title":"Easy Sequences 66: Fractional Part of Sum of Consecutive Powers over a Prime","description":"Given three postive integers, ,  and , create the function , defined as follows:\r\n            \r\nWhere  is the th prime and the symbol  is the  fractional part function of a positive real number, defined as follows: \r\n        If  then , where , is the 'floor' of .\r\nFor example for ,  and , we have:\r\n        \r\nTherefore, since :\r\n        \r\nPlease round-off the output to four decimal places, so in this case: .\r\nHINT: This is related to the Faulhaber's formula.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 355px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 177.5px; transform-origin: 407px 177.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 106px 8px; transform-origin: 106px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven three postive integers, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ee\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17px 8px; transform-origin: 17px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73px 8px; transform-origin: 73px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, create the function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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style=\"width: 85px; height: 18.5px;\" width=\"85\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62px 8px; transform-origin: 62px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, defined as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e            \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 181px; height: 45px;\" width=\"181\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.5px 8px; transform-origin: 77.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime and the symbol \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 18px; height: 18.5px;\" width=\"18\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the  \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://brilliant.org/wiki/factional-part-function/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003efractional part function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.5px 8px; transform-origin: 143.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a positive real number, defined as follows: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.5px 8px; transform-origin: 21.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 46px; height: 19px;\" width=\"46\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e then \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 86.5px; height: 18.5px;\" width=\"86.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25px 8px; transform-origin: 25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 23px; height: 18.5px;\" width=\"23\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50px 8px; transform-origin: 50px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is the 'floor' of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.5px 8px; transform-origin: 51.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 44px; height: 18px;\" width=\"44\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEgAAAAkCAYAAAAq23xmAAACq0lEQVRoQ+2YuUrEQBzG3QcQPDpBEK2sFLwaFRREsdVCwQdQCxtBQS1VUKwsFEHBTi1sxaMQPArxQCsL0TfwegL9PpiR2exMZncnS9zsBD6ySWYm+f/yv7KpMr+FEkh5PuEEPCCLh3hAHpBbEvEe5D0ojUA9jmrEmWfs323+VSoeNAgQ61BDAMgJjiehNxOoUgC0BOPnQjzlE9daTZCSDqgThl9Cm9CagFCNfR+0qHjUHX636SAmHdAtjKYYRsGNoF6gSnGhGfun4KAkA2qCsYdQB2RKxmr4jWPcVikBYmKu1RmtQJAhyFORAKJbNoobXOlitsjOqYC68OwZNmUbYiOYPC2MP8WeVeEVGtLFrQESXb48AoBRvhjatSds0YaiDRAbqwOIZXAU2hcGHmPfDy1D81kaLedkOVw7jCW5ymWBwNwNHE9As9Cqbt0wQHzj5xCzfDA+Pwznw56dCbHF0bgvzOdbj2KTVYy2GBO5CRAn30DsPOlB6kPN4HgFivptRmF0LmtIO7S5Ry5kAiRdj+PU/kDGLOH0QBl9Qy5PGONYRscjpKYN7ePoAMnJnCC9h9l+AWLeYbzuQNYPvRgBhN1aRgdts+ZPHSDVe9iCMwexcl1AZw5g/kMVIxwWC9pjhUPKOkBszVm1os4xcVexnOGYAP0I/zR+wOUZOnFXMb54m+ew+z5S7dN5kCzhBDRgCCkayzxk/B8lT4iFmsa0wZ6O9pg2FqBuKO3D1paDgv0P88i2UMaHXaGsc1xX5lQ2td+Gtepwng1jRsnXASJpmZy5HrP9A9QLtUNjQTd0NKCQ09WCY7uPNqWENYrDWJG9TgV0D10XERjCYD6ZslFRru/it/yU+jtt+xbLYf1kDvWALO/VA/KA3ELfe5D3IO9BbgS8B7nx8znIwu8XGuN3JcauM6EAAAAASUVORK5CYII=\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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src=\"data:image/png;base64,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\" style=\"width: 49px; height: 20px;\" width=\"49\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 35px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.5px; text-align: left; transform-origin: 384px 17.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; 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\" style=\"width: 430.5px; height: 35px;\" width=\"430.5\" height=\"35\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 210.5px 8px; transform-origin: 210.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ePlease round-off the output to four decimal places, so in this case: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 152px; height: 18.5px;\" width=\"152\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17px 8px; transform-origin: 17px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHINT:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.5px 8px; transform-origin: 68.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e This is related to the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Faulhaber%27s_formula\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eFaulhaber's formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = FPSP(x,e,n)\r\n    y = FPSP(x,e,n);\r\nend","test_suite":"%%\r\nx = 10; e = 2; n = 6;\r\ny_correct = 0.6154;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 100; e = 3; n = 50;\r\ny_correct = 0.6288;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 100; e = 4; n = 100;\r\ny_correct = 0.2495;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 250; e = 5; n = 200;\r\ny_correct = 0.1415;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 123; e = 6; n = 200;\r\ny_correct = 0.1292;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 10; e = 10; n = 300;\r\ny_correct = 0.7006;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 1234; e = 56; n = 789;\r\ny_correct = 0.8419;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nx = 3000; e = 300; n = 3000;\r\ny_correct = 0.6441;\r\nassert(isequal(FPSP(x,e,n),y_correct))\r\n%%\r\nxs = 20:2:2000;\r\nys = arrayfun(@(x) FPSP(x,x/2,x),xs);\r\nss = round([sum(ys) nnz(ys) mean(ys) mode(ys) median(ys) std(ys)],4); \r\nss_correct = [472.0982 numel(xs) 0.4764 0.2518 0.4704 0.2854];\r\nassert(isequal(ss,ss_correct))\r\n%%\r\nfiletext = fileread('FPSP.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'BigInteger') || contains(filetext, 'java'); \r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":"2022-01-20T12:19:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-01-19T18:18:26.000Z","updated_at":"2026-05-24T23:36:48.000Z","published_at":"2022-01-20T10:08:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven three postive integers, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, create the function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eFPSP(x,y,n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, defined as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e            \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eFPSP(x,e,n)=\\\\left\\\\{ \\\\frac{1}{p_{n}} \\\\cdot \\\\sum^{x}_{i=1} i^{e}\\\\right\\\\}  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_{n}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime and the symbol \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\{\\\\}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://brilliant.org/wiki/factional-part-function/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003efractional part function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e of a positive real number, defined as follows: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er \\\\in \\\\mathbb{R}^{+}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\{r\\\\}=r-\\\\lfloor r \\\\rfloor\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lfloor r \\\\rfloor\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is the 'floor' of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee=2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en=6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sum^{10}_{i=1} i^{2} = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTherefore, since \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_6=13\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eFPSP(10,2,6)=\\\\left\\\\{ \\\\frac{385}{13} \\\\right\\\\}  = \\\\{29.615384561538...\\\\}=0.61538461538...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease round-off the output to four decimal places, so in this case: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eFPSP(10,2,6)=0.6154\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHINT:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e This is related to the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Faulhaber%27s_formula\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eFaulhaber's formula\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":61052,"title":"Compute the Dedekind psi function","description":"Write a function to compute the Dedekind psi function. The values for n = 1, 2, 3, 4, 5, and 6 are 1, 3, 4, 6, 6, and 12. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 21px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 10.5px; transform-origin: 408px 10.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/DedekindFunction.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eDedekind psi function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The values for n = 1, 2, 3, 4, 5, and 6 are 1, 3, 4, 6, 6, and 12. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = DedekindPsi(n)\r\n  y = psi(n);\r\nend","test_suite":"%%\r\nn = 1;\r\ny = DedekindPsi(n);\r\ny_correct = 1;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 8;\r\ny = DedekindPsi(n);\r\ny_correct = 12;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 49;\r\ny = DedekindPsi(n);\r\ny_correct = 56;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 312;\r\ny = DedekindPsi(n);\r\ny_correct = 672;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 1441;\r\ny = DedekindPsi(n);\r\ny_correct = 1584;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 8028;\r\ny = DedekindPsi(n);\r\ny_correct = 16128;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 14035;\r\ny = DedekindPsi(n);\r\ny_correct = 19296;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 85432;\r\ny = DedekindPsi(n);\r\ny_correct = 131040;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 6645344;\r\ny = DedekindPsi(n);\r\ny_correct = 10402560;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 5533221188;\r\ny = DedekindPsi(n);\r\ny_correct = 9083498112;\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\np = primes(1e8);\r\nq = p(randi(length(p)));\r\ny = DedekindPsi(q);\r\nassert(isequal(y,q+1))\r\n\r\n%%\r\np = primes(200);\r\nq = p(randi(length(p)));\r\nn = randi(5);\r\nx = q^n;\r\ny = DedekindPsi(x);\r\nassert(isequal(y,x+x/q))\r\n\r\n%%\r\np = primes(200);\r\nk = unique(randi(length(p),[1 randi(5)]));\r\nq = p(k);\r\nx = prod(q);\r\ny = DedekindPsi(x);\r\nassert(isequal(y,prod(q+1)))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2025-10-25T15:14:46.000Z","deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-10-25T15:09:34.000Z","updated_at":"2026-06-06T15:56:45.000Z","published_at":"2025-10-25T15:14:46.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/DedekindFunction.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDedekind psi function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. The values for n = 1, 2, 3, 4, 5, and 6 are 1, 3, 4, 6, 6, and 12. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1927,"title":"Find nth maximum","description":"Find nth maximum in a vector of integer numbers. Return NaN if no such number exists.\r\n\r\n x = [2 6 4 9 -10 3 1 5 -10];\r\n\r\nSo \r\n\r\n* n_max(x,3) would find 3rd maximum value in x which is 5\r\n* n_max(x,8) would find 8th maximum value in x which is -10\r\n* n_max(x,9) would find 9th maximum value in x which is -10\r\n* n_max(x,12) does not exist so return NaN\r\n\r\n","description_html":"\u003cp\u003eFind nth maximum in a vector of integer numbers. Return NaN if no such number exists.\u003c/p\u003e\u003cpre\u003e x = [2 6 4 9 -10 3 1 5 -10];\u003c/pre\u003e\u003cp\u003eSo\u003c/p\u003e\u003cul\u003e\u003cli\u003en_max(x,3) would find 3rd maximum value in x which is 5\u003c/li\u003e\u003cli\u003en_max(x,8) would find 8th maximum value in x which is -10\u003c/li\u003e\u003cli\u003en_max(x,9) would find 9th maximum value in x which is -10\u003c/li\u003e\u003cli\u003en_max(x,12) does not exist so return NaN\u003c/li\u003e\u003c/ul\u003e","function_template":"function y = n_max(x,n)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [2 6 4 9 -10 3 1 5 -10];\r\ny_correct = 5;\r\nassert(isequaln(n_max(x,3),y_correct))\r\n\r\n%%\r\nx = [2 6 4 9 -10 3 1 5 -10];\r\ny_correct = NaN;\r\nassert(isequaln(n_max(x,13),y_correct))\r\n\r\n%%\r\nx = [0 0 0 -2 0 -6];\r\ny_correct = -2;\r\nassert(isequaln(n_max(x,5),y_correct))\r\n\r\n%%\r\nx = [1 1 1 1 1 1];\r\ny_correct = 1;\r\nassert(isequaln(n_max(x,4),y_correct))\r\n\r\n%%\r\nx = [1 1 1 1 1 1];\r\ny_correct = NaN;\r\nassert(isequaln(n_max(x,14),y_correct))","published":true,"deleted":false,"likes_count":6,"comments_count":4,"created_by":18172,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":942,"test_suite_updated_at":"2013-10-15T19:44:56.000Z","rescore_all_solutions":false,"group_id":13,"created_at":"2013-10-10T06:23:03.000Z","updated_at":"2026-07-19T12:00:35.000Z","published_at":"2013-10-10T06:28:51.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind nth maximum in a vector of integer numbers. Return NaN if no such number exists.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [2 6 4 9 -10 3 1 5 -10];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSo\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en_max(x,3) would find 3rd maximum value in x which is 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en_max(x,8) would find 8th maximum value in x which is -10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en_max(x,9) would find 9th maximum value in x which is -10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003en_max(x,12) does not exist so return NaN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60977,"title":"Linear primes p' = mp + n","description":"Problem statement\r\n\r\nList the prime numbers of the form p' = mp + n for a given (m,n) couple and such that p' \u003c k, k positive integer.\r\n\r\nExamples \r\n\r\n(m,n) = (3,2) and k = 50   =\u003e p = [11, 17, 23, 41];\r\n(m,n) = (4,3) and k = 50   =\u003e p = [11, 23, 31, 47];\r\n(m,n) = (5,4) and k = 100 =\u003e p = [19, 29, 59, 89];\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 536.033px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 268.017px; transform-origin: 408px 268.017px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.508px 8px; transform-origin: 108.508px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eList the prime numbers of the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.7917px 8px; transform-origin: 34.7917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep' = mp + n\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5583px 8px; transform-origin: 36.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for a given \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.3333px 8px; transform-origin: 16.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(m,n)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.4583px 8px; transform-origin: 68.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e couple and such that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.025px 8px; transform-origin: 26.025px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep' \u0026lt; k, k \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.625px 8px; transform-origin: 48.625px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003epositive integer.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 149.425px 8px; transform-origin: 149.425px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(m,n) = (3,2) and k = 50   =\u0026gt; p = [11, 17, 23, 41];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 149.425px 8px; transform-origin: 149.425px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(m,n) = (4,3) and k = 50   =\u0026gt; p = [11, 23, 31, 47];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 149.95px 8px; transform-origin: 149.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(m,n) = (5,4) and k = 100 =\u0026gt; p = [19, 29, 59, 89];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = list_the_linear_primes(m,n,k)\r\n  p = m;\r\nend","test_suite":"%%\r\nm = 3;\r\nn = 2;\r\nk = 50;\r\np_correct = [11, 17, 23, 41];\r\nassert(isequal(list_the_linear_primes(m,n,k),p_correct))\r\n\r\n%%\r\nm = 4;\r\nn = 3;\r\nk = 50;\r\np_correct = [11, 23, 31, 47];\r\nassert(isequal(list_the_linear_primes(m,n,k),p_correct))\r\n\r\n%%\r\nm = 5;\r\nn = 4;\r\nk = 100;\r\np_correct = [19, 29, 59, 89];\r\nassert(isequal(list_the_linear_primes(m,n,k),p_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('list_the_linear_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-21T10:46:17.000Z","deleted_by":null,"deleted_at":null,"solvers_count":37,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-21T10:27:56.000Z","updated_at":"2026-07-20T06:35:34.000Z","published_at":"2025-07-21T10:46:17.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eList the prime numbers of the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep' = mp + n\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for a given \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(m,n)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e couple and such that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep' \u0026lt; k, k \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003epositive integer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(m,n) = (3,2) and k = 50   =\u0026gt; p = [11, 17, 23, 41];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(m,n) = (4,3) and k = 50   =\u0026gt; p = [11, 23, 31, 47];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(m,n) = (5,4) and k = 100 =\u0026gt; p = [19, 29, 59, 89];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52921,"title":"Easy Sequences 40: Quadratic Congruence","description":"Quadratic Congruence is a modular equation of the form: . \r\nIn this exercise you will be given a vector containing the coefficients of a quadratic polynomial (), and a modulus base (). Using these data, create a function that outputs the pair (), which are the 'primitive' solutions to the quadratic congruence.\r\nFor example consider the congruence: ,  the solution is , since:\r\n                                    , and\r\n                                    .\r\nNOTE: A primitive modulus to base , can only have values from  to . This is a simplified problem, in which the quadratic polynomials given in the test suite, are all factorable, and the modulus base are all odd primes.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 235px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 117.5px; transform-origin: 407px 117.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297px 8px; transform-origin: 297px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn this exercise you will be given a vector containing the coefficients of a quadratic polynomial (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 79px; height: 18.5px;\" width=\"79\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.5px 8px; transform-origin: 26.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e), and a modulus base (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.5px 8px; transform-origin: 6.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e). \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 200.5px 8px; transform-origin: 200.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eUsing these data, create a function that outputs the pair (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 75px; height: 20px;\" width=\"75\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 56px 8px; transform-origin: 56px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e), which are the 'primitive' solutions to the quadratic congruence.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 123.5px 8px; transform-origin: 123.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example consider the congruence: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 157.5px; height: 19.5px;\" width=\"157.5\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.5px 8px; transform-origin: 51.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e,  the solution is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 35px; height: 18.5px;\" width=\"35\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.5px 8px; transform-origin: 22.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, since:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 292px; height: 19.5px;\" width=\"292\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 292px; height: 19.5px;\" width=\"292\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNOTE: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.5px 8px; transform-origin: 88.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA primitive modulus to base \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.5px 8px; transform-origin: 88.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, can only have values from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003e0\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10px 8px; transform-origin: 10px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 38.5px; height: 18px;\" width=\"38.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132px 8px; transform-origin: 132px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. This is a simplified problem, in which the quadratic polynomials given in the test suite, are all factorable, and the modulus base are all odd primes.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function xs = quadCong(cs,m)\r\n    xs = cs;\r\nend","test_suite":"%%\r\ncs = [1,5,6]; m = 3;\r\nxs_correct = [0,1];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [1,5,6]; m = 7;\r\nxs_correct = [4,5];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [6,-7,-5]; m = 11;\r\nxs_correct = [5,9];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [35,76,33]; ms = setdiff(primes(10000),primes(100));\r\nys = arrayfun(@(m) sum(quadCong(cs,m)),ms);\r\nss = [sum(ys) floor([mean(ys) mode(ys) median(ys) std(ys)])];\r\nss_correct = [5752615 4777 57 4151 3465];\r\nassert(isequal(ss,ss_correct))\r\n%%\r\ncs = [10,-131,391]; m = 6373;\r\nxs_correct = [3195,5103];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [253,-1420,672]; m = 30103;\r\nxs_correct = [3927,8215];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [528,26711,283500]; m = 198733;\r\nxs_correct = [37227,174629];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\ncs = [4545,13092,-69741]; m = 1000121;\r\nxs_correct = [821876,933449];\r\nassert(isequal(quadCong(cs,m),xs_correct))\r\n%%\r\nfiletext = fileread('quadCong.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'BigInteger') || contains(filetext, 'java'); \r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":19,"test_suite_updated_at":"2021-10-17T05:04:28.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-10-12T12:11:53.000Z","updated_at":"2026-05-24T20:40:09.000Z","published_at":"2021-10-16T17:40:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eQuadratic Congruence is a modular equation of the form: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eax^{2}+bx+c\\\\equiv 0\\\\  (mod\\\\  m)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn this exercise you will be given a vector containing the coefficients of a quadratic polynomial (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ecs =\\\\left[ a,b,c\\\\right]  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e), and a modulus base (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e). \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eUsing these data, create a function that outputs the pair (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003exs=[x_{1},x_{2}]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e), which are the 'primitive' solutions to the quadratic congruence.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example consider the congruence: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^{2}+5x+6\\\\equiv 0\\\\  (mod\\\\  7)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,  the solution is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e[4,5]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, since:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e4^{2}+5\\\\cdot 4+6= 16+20+6= 42\\\\equiv 0\\\\  (mod\\\\  7)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e5^{2}+5\\\\cdot 5+6= 25+25+6= 56\\\\equiv 0\\\\  (mod\\\\  7)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNOTE: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eA primitive modulus to base \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, can only have values from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em-1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. This is a simplified problem, in which the quadratic polynomials given in the test suite, are all factorable, and the modulus base are all odd primes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44068,"title":"The number of trailing zero digit of a factorial","description":"For given positive integer n, take factorial of that number. How many trailing zeros does it have?\r\n\r\nExample: factorial(11) = 39916800\r\n\r\nIts last zero-digit count is 2.\r\n\r\nOptional: Can you make an efficient algorithm for a very large n?","description_html":"\u003cp\u003eFor given positive integer n, take factorial of that number. How many trailing zeros does it have?\u003c/p\u003e\u003cp\u003eExample: factorial(11) = 39916800\u003c/p\u003e\u003cp\u003eIts last zero-digit count is 2.\u003c/p\u003e\u003cp\u003eOptional: Can you make an efficient algorithm for a very large n?\u003c/p\u003e","function_template":"function ct = powerTenInFactorial(n)\r\n  ct = 0;\r\nend","test_suite":"%%\r\nn = 1;\r\nct_correct = 0;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))\r\n\r\n%%\r\nn = 9;\r\nct_correct = 1;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))\r\n\r\n%%\r\nn = 27;\r\nct_correct = 6;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))\r\n\r\n%%\r\nn = 626;\r\nct_correct = 156;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))\r\n\r\n%%\r\nn = 620;\r\nct_correct = 152;\r\nassert(isequal(powerTenInFactorial(n),ct_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":3,"created_by":115733,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":66,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":673,"created_at":"2017-02-14T00:24:29.000Z","updated_at":"2026-03-20T13:50:01.000Z","published_at":"2017-02-14T00:24:29.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor given positive integer n, take factorial of that number. How many trailing zeros does it have?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample: factorial(11) = 39916800\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIts last zero-digit count is 2.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOptional: Can you make an efficient algorithm for a very large n?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1771,"title":"Polygonal numbers","description":"The task of \u003chttp://www.mathworks.co.uk/matlabcentral/cody/problems/5 Problem 5\u003e is to calculate triangular numbers. By playing with dots we can produce also square numbers like:\r\n\r\n                                 * * * *\r\n                     * * *       * * * *\r\n           * *       * * *       * * * *\r\n1: *    4: * *    9: * * *   16: * * * *\r\n\r\nor hexagonal numbers:\r\n\r\n                                          * * * *\r\n                                         *       *\r\n                         * * *          * * * *   *\r\n                        *     *        * *     *   *\r\n            * *        * * *   *        * * *   * * \r\n           *   *        *   * *          *   * * *\r\n1: *    6:  * *    15:   * * *       28:  * * * *\r\n\r\n\u0026nbsp\r\n\r\nAccording to those rules we can create \u003chttp://en.wikipedia.org/wiki/Polygonal_number polygonal numbers\u003e for all regular polygons.\r\n\r\n\u0026nbsp\r\n\r\nYour task: given _S_ and _N_ *calculate _N_-th _S_-gonal numbers* _P(S,N)_ \r\n\r\n\u0026nbsp\r\n\r\nExamples:\r\n\r\n# _P(4, 3)_ returns _[9]_    because 3-rd square number is 9,\r\n# _P(3, 1:5)_ returns _[1, 3, 6, 10, 15]_   first 5 triangular numbers,\r\n# _P(3:6, 4)_ returns _[10, 16, 22, 28]_   4-th triangular, square, pentagonal and hexagonal numbers,\r\n# _P([3, 4], [1; 2])_ returns _[1, 1; 3, 4]_.\r\n\r\nsee the test suite for more hints","description_html":"\u003cp\u003eThe task of \u003ca href = \"http://www.mathworks.co.uk/matlabcentral/cody/problems/5\"\u003eProblem 5\u003c/a\u003e is to calculate triangular numbers. By playing with dots we can produce also square numbers like:\u003c/p\u003e\u003cpre\u003e                                 * * * *\r\n                     * * *       * * * *\r\n           * *       * * *       * * * *\r\n1: *    4: * *    9: * * *   16: * * * *\u003c/pre\u003e\u003cp\u003eor hexagonal numbers:\u003c/p\u003e\u003cpre\u003e                                          * * * *\r\n                                         *       *\r\n                         * * *          * * * *   *\r\n                        *     *        * *     *   *\r\n            * *        * * *   *        * * *   * * \r\n           *   *        *   * *          *   * * *\r\n1: *    6:  * *    15:   * * *       28:  * * * *\u003c/pre\u003e\u003cp\u003e\u0026nbsp\u003c/p\u003e\u003cp\u003eAccording to those rules we can create \u003ca href = \"http://en.wikipedia.org/wiki/Polygonal_number\"\u003epolygonal numbers\u003c/a\u003e for all regular polygons.\u003c/p\u003e\u003cp\u003e\u0026nbsp\u003c/p\u003e\u003cp\u003eYour task: given \u003ci\u003eS\u003c/i\u003e and \u003ci\u003eN\u003c/i\u003e \u003cb\u003ecalculate \u003ci\u003eN\u003c/i\u003e-th \u003ci\u003eS\u003c/i\u003e-gonal numbers\u003c/b\u003e \u003ci\u003eP(S,N)\u003c/i\u003e\u003c/p\u003e\u003cp\u003e\u0026nbsp\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003col\u003e\u003cli\u003e\u003ci\u003eP(4, 3)\u003c/i\u003e returns \u003ci\u003e[9]\u003c/i\u003e    because 3-rd square number is 9,\u003c/li\u003e\u003cli\u003e\u003ci\u003eP(3, 1:5)\u003c/i\u003e returns \u003ci\u003e[1, 3, 6, 10, 15]\u003c/i\u003e   first 5 triangular numbers,\u003c/li\u003e\u003cli\u003e\u003ci\u003eP(3:6, 4)\u003c/i\u003e returns \u003ci\u003e[10, 16, 22, 28]\u003c/i\u003e   4-th triangular, square, pentagonal and hexagonal numbers,\u003c/li\u003e\u003cli\u003e\u003ci\u003eP([3, 4], [1; 2])\u003c/i\u003e returns \u003ci\u003e[1, 1; 3, 4]\u003c/i\u003e.\u003c/li\u003e\u003c/ol\u003e\u003cp\u003esee the test suite for more hints\u003c/p\u003e","function_template":"function y = P(s,n)\r\n  y = s*n;\r\nend","test_suite":"%%\r\nassert(P(4,3)==9)\r\n%%\r\ns=3;\r\nn=1:5;\r\ny_correct=[1 3 6 10 15];\r\nassert(isequal(P(s,n),y_correct))\r\n%%\r\ns=3:6;\r\nn=4;\r\nY_correct=[10 16 22 28];\r\nassert(isequal(P(s,n),Y_correct))\r\n%%\r\ns=randi([3 1000],1,10);\r\nn=1;\r\ny_correct=ones(1,10);\r\nassert(isequal(P(s,n),y_correct))\r\n%%\r\ns=randi(1000,1,10)+2;\r\nn=5;\r\ny_correct=10*s-15;\r\nassert(isequal(P(s,n),y_correct))\r\n%%\r\ns=[5 6 7]';\r\nn=[2 8 9];\r\ny_correct=[5 92 117;6 120 153;7 148 189];\r\nassert(isequal(P(s,n),y_correct))\r\n%%\r\n% for M.Z.\r\ns=[5 6 7]';\r\nn=[2 8 9];\r\ny_correct=[5 92 117;6 120 153;7 148 189];\r\nassert(isequal(P(flipud(s),n),flipud(y_correct)))\r\n%%\r\ns=randi([3,1000]);\r\nn=randi([3,1000]);\r\nassert(P(s+1,n+1)-P(s,n+1)==P(3,n))\r\n","published":true,"deleted":false,"likes_count":5,"comments_count":4,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":140,"test_suite_updated_at":"2016-10-15T05:56:15.000Z","rescore_all_solutions":true,"group_id":8,"created_at":"2013-08-02T13:16:29.000Z","updated_at":"2026-07-15T12:56:22.000Z","published_at":"2013-08-02T23:34:24.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe task of\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.co.uk/matlabcentral/cody/problems/5\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is to calculate triangular numbers. By playing with dots we can produce also square numbers like:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[                                 * * * *\\n                     * * *       * * * *\\n           * *       * * *       * * * *\\n1: *    4: * *    9: * * *   16: * * * *]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eor hexagonal numbers:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[                                          * * * *\\n                                         *       *\\n                         * * *          * * * *   *\\n                        *     *        * *     *   *\\n            * *        * * *   *        * * *   * * \\n           *   *        *   * *          *   * * *\\n1: *    6:  * *    15:   * * *       28:  * * * *]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026amp;nbsp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAccording to those rules we can create\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Polygonal_number\\\"\u003e\u003cw:r\u003e\u003cw:t\u003epolygonal numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for all regular polygons.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026amp;nbsp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task: given\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ecalculate\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e-th\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e-gonal numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP(S,N)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026amp;nbsp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP(4, 3)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[9]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e because 3-rd square number is 9,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP(3, 1:5)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[1, 3, 6, 10, 15]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e first 5 triangular numbers,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP(3:6, 4)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[10, 16, 22, 28]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e 4-th triangular, square, pentagonal and hexagonal numbers,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP([3, 4], [1; 2])\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e returns\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[1, 1; 3, 4]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003esee the test suite for more hints\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46054,"title":"Count trailing zeros in a primorial","description":"\u003chttps://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial Cody Problem 44068\u003e asked us to count the  trailing zeros in a factorial. This problem deals with the  _\u003chttps://mathworld.wolfram.com/Primorial.html primorial\u003e_. If p_n is the nth prime number, then the primorial p_n# is the product of the prime numbers up to and including p_n. For example, if n = 5, then p_5# = 2*3*5*7*11 = 2310, which has one trailing zero. \r\n\r\nCount the trailing zeros in the primorial p_n#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.* \r\n\r\nPlease make your code general because I might add more tests later. \r\n\r\n*;-)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 205px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 102.5px; transform-origin: 407px 102.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.8833px; text-align: left; transform-origin: 384px 31.8833px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 44068\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 238.833px 7.91667px; transform-origin: 238.833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asked us to count the trailing zeros in a factorial. This problem deals with the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/Primorial.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eprimorial\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 7.91667px; transform-origin: 9.71667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6167px 7.91667px; transform-origin: 20.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.28333px 7.91667px; transform-origin: 7.28333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime number, then the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 178.55px 7.91667px; transform-origin: 178.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# is the product of the prime numbers up to and including \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 52.1167px 7.91667px; transform-origin: 52.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.85px 7.91667px; transform-origin: 29.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 5, then \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_5\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.8667px 7.91667px; transform-origin: 11.8667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2*3*5*7*11\" style=\"width: 92px; height: 18px;\" width=\"92\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 112.233px 7.91667px; transform-origin: 112.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 2310, which has one trailing zero.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.1333px; text-align: left; transform-origin: 384px 21.1333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 121.767px 7.91667px; transform-origin: 121.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCount the trailing zeros in the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.467px 7.91667px; transform-origin: 254.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.1px 7.91667px; transform-origin: 215.1px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ePlease make your code general because I might add more tests later.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problems 46057 \"Find the last non-zero digit in a primorial\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.91667px; transform-origin: 15.5667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46060 \"Identify primorial primes\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.33333px 7.91667px; transform-origin: 9.33333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e*;-)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = primorialTrailingZeros(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = 0;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 2;\r\ny_correct = 0;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 5;\r\ny_correct = 1;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 10;\r\ny_correct = sin((n+3)*pi/2);\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 100;\r\ny_correct = sum(1./2.^(1:n));\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 1e4;\r\ny_correct = besselj(mod(n,10),prod(num2str(n)'-'0'));\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 1e6;\r\nf = [factor(log10(n)) 3 7 7 89];\r\ny_correct = length(primes(n))/prod(f);\r\nassert(isequal(primorialTrailingZeros(n),y_correct))","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":84,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-26T15:54:06.000Z","updated_at":"2026-05-22T14:07:45.000Z","published_at":"2020-07-26T16:20:27.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 44068\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asked us to count the trailing zeros in a factorial. This problem deals with the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/Primorial.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprimorial\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime number, then the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# is the product of the prime numbers up to and including \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 5, then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2*3*5*7*11\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\cdot3\\\\cdot5\\\\cdot7\\\\cdot11\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 2310, which has one trailing zero.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCount the trailing zeros in the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease make your code general because I might add more tests later.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems 46057 \\\"Find the last non-zero digit in a primorial\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46060 \\\"Identify primorial primes\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e*;-)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60943,"title":"Ulam primes second formula","description":"Historical context\r\n\r\n\r\nThe polish-american mathematician Stanislaw Ulam found the formula  to give some prime numbers not given neither by Euler's formula neither by its own first formula. \r\n\r\n\r\nProblem statement\r\n\r\nFor a given integer m, m \u003e= 59, list the Ulam prime numbers less or equal to m in a vector up. Also compute their frequency / ratio f compare to all the primes in the same range. If m is not a positive integer greater or equal to 59, your function should return up = [] and f = 0.\r\n\r\nExamples (check the tests for more)\r\n\r\nIf m = 59 then up = 59 and f = 1 since 59 = P(0) and it is the only prime number m greater or equal and less or equal to 59;\r\nIf m = 200 then up = [59, 67, 83, 107, 139, 179] and f = 6/30 since 59 = P(0), 67 = P(1), 83 = P(2), 107 = P(3), 139 = P(4), 179 = P(5), and there are in total 30 prime numbers in the range [59; 200] ; \r\nIf m = 23 then up = [] and f = 0;\r\n\r\n\r\nTip \r\n\r\n here.\r\n\r\n\r\nFobidden functions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\n\r\n\r\nSee also\r\nUlam primes first formula\r\nEuler primes\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1069.47px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 534.733px; transform-origin: 408px 534.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 219.4px 8px; transform-origin: 219.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe polish-american mathematician Stanislaw Ulam found the formula \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"140\" height=\"19.5\" style=\"width: 140px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.1833px 8px; transform-origin: 92.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to give some prime numbers not given neither by Euler's formula neither by its own first formula. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.075px 8px; transform-origin: 26.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u0026gt;= 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 138.475px 8px; transform-origin: 138.475px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Ulam prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0583px 8px; transform-origin: 40.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector u\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.3833px 8px; transform-origin: 96.3833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 294.442px 8px; transform-origin: 294.442px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.35px 8px; transform-origin: 63.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.075px 8px; transform-origin: 25.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e up = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3333px 8px; transform-origin: 79.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests for more)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 51.0833px; transform-origin: 392px 51.0833px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6.21667px 8px; transform-origin: 6.21667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e u\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.0333px 8px; transform-origin: 20.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.1417px 8px; transform-origin: 16.1417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.0667px 8px; transform-origin: 26.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.2px 8px; transform-origin: 21.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e = P(0)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 99.575px 8px; transform-origin: 99.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and it is the only prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.533px 8px; transform-origin: 108.533px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e greater or equal and less or equal to 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2.33333px 8px; transform-origin: 2.33333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 200\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6.21667px 8px; transform-origin: 6.21667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e u\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 94.7083px 8px; transform-origin: 94.7083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [59, 67, 83, 107, 139, 179]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 6/30 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.0667px 8px; transform-origin: 26.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince 59\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 147.225px 8px; transform-origin: 147.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e = P(0), 67 = P(1), 83 = P(2), 107 = P(3), 139 = P(4), 179 = P(5), \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 73.9083px 8px; transform-origin: 73.9083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand there are in total 30\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.9917px 8px; transform-origin: 34.9917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [59; 200] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 23.925px 8px; transform-origin: 23.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 23 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.45px 8px; transform-origin: 19.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen u\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.9167px 8px; transform-origin: 16.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630/problems/60944\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eUlam primes first formula\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60942-euler-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eEuler primes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [up,f] = ulam_primes2(m)\r\n  up = m;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 59;\r\nup_correct = 59;\r\nf_correct = 1;\r\n[up,f] = ulam_primes2(m);\r\nassert(isequal([up,f],[up_correct,f_correct]));\r\n\r\n%%\r\nm = 200;\r\nup_correct = [59, 67, 83, 107, 139, 179];\r\nf_correct = 1/5;\r\n[up,f] = ulam_primes2(m);\r\nassert(isequal([up,f],[up_correct,f_correct]));\r\n\r\n%%\r\nm = 500;\r\nup_correct = [59, 67, 83, 107, 139, 179, 227, 283, 347, 419, 499];\r\nf_correct = 11/79;\r\n[up,f] = ulam_primes2(m);\r\nassert(isequal([up,f],[up_correct,f_correct]));\r\n\r\n%%\r\nm = 23;\r\nup_correct = [];\r\nf_correct = 0;\r\n[up,f] = ulam_primes2(m);\r\nassert(isequal([up,f],[up_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('ulam_primes2.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:08:26.000Z","deleted_by":null,"deleted_at":null,"solvers_count":22,"test_suite_updated_at":"2025-07-09T05:56:45.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-26T19:08:27.000Z","updated_at":"2026-06-09T14:26:47.000Z","published_at":"2025-06-26T19:44:55.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe polish-american mathematician Stanislaw Ulam found the formula \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{P(n) = 4n^2 + 4n + 59}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to give some prime numbers not given neither by Euler's formula neither by its own first formula. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u0026gt;= 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Ulam prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector u\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If m is not a positive integer greater or equal to 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e up = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e u\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e = P(0)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and it is the only prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e greater or equal and less or equal to 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 200\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e u\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [59, 67, 83, 107, 139, 179]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 6/30 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince 59\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e = P(0), 67 = P(1), 83 = P(2), 107 = P(3), 139 = P(4), 179 = P(5), \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand there are in total 30\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [59; 200] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 23 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen u\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":49582,"title":"Determine whether a square can be drawn on a grid of dots","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 444.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 222.233px; transform-origin: 407px 222.233px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.358px 7.79167px; transform-origin: 382.358px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSuppose you have a regular grid of dots, each 1 unit apart. By connecting four dots arranged in a square of side 1, you can draw a square of area 1. Similarly, you can draw squares of area 4, 9, 16, 25, etc. As shown below, by connecting dots with diagonal lines, you can draw squares of area 2, 5, 10, 8, 13, and 17. Which squares can you draw by connecting dots on the grid with straight lines? \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 301.708px 7.79167px; transform-origin: 301.708px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes an area and determines whether the square of that area can be drawn.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 7.79167px; transform-origin: 0px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 291.467px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 145.733px; text-align: left; transform-origin: 384px 145.733px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://www.mathsisfun.com/activity/images/squares8.gif\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = squareOnGrid(A)\r\n  tf = false;\r\nend","test_suite":"%%\r\nassert(squareOnGrid(1))\r\n\r\n%%\r\nassert(squareOnGrid(2))\r\n\r\n%%\r\nassert(~squareOnGrid(3))\r\n\r\n%%\r\nassert(squareOnGrid(4))\r\n\r\n%%\r\nassert(squareOnGrid(5))\r\n\r\n%%\r\nassert(~squareOnGrid(6))\r\n\r\n%%\r\nassert(~squareOnGrid(7))\r\n\r\n%%\r\nassert(squareOnGrid(8))\r\n\r\n%%\r\nassert(squareOnGrid(9))\r\n\r\n%%\r\nassert(squareOnGrid(10))\r\n\r\n%%\r\nassert(~squareOnGrid(11))\r\n\r\n%%\r\nassert(~squareOnGrid(12))\r\n\r\n%%\r\nassert(squareOnGrid(13))\r\n\r\n%%\r\nassert(~squareOnGrid(14))\r\n\r\n%%\r\nassert(~squareOnGrid(15))\r\n\r\n%%\r\nassert(squareOnGrid(16))\r\n\r\n%%\r\nassert(squareOnGrid(17))\r\n\r\n%%\r\nassert(squareOnGrid(18))\r\n\r\n%%\r\nassert(~squareOnGrid(19))\r\n\r\n%%\r\nassert(squareOnGrid(20))\r\n\r\n%%\r\nassert(~squareOnGrid(21))\r\n\r\n%%\r\nassert(~squareOnGrid(42))\r\n\r\n%%\r\nassert(squareOnGrid(49))\r\n\r\n%%\r\nassert(~squareOnGrid(51))\r\n\r\n%%\r\nassert(squareOnGrid(53))\r\n\r\n%%\r\nassert(~squareOnGrid(63))\r\n\r\n%%\r\nassert(squareOnGrid(130))\r\n\r\n%%\r\nassert(squareOnGrid(281))\r\n\r\n%%\r\nassert(~squareOnGrid(283))\r\n\r\n%%\r\nfor k = 566:575\r\n    tf(k-565) = squareOnGrid(k);\r\nend\r\nassert(isequal(sum(tf),1))\r\n\r\n%%\r\nassert(~squareOnGrid(1323))\r\n\r\n%%\r\nassert(~squareOnGrid(2310))\r\n\r\n%%\r\nassert(~squareOnGrid(12768))\r\n\r\n%%\r\nassert(squareOnGrid(12769))\r\n\r\n%%\r\nassert(~squareOnGrid(19200))\r\n\r\n%%\r\nassert(~squareOnGrid(1514205))\r\n\r\n%%\r\nassert(squareOnGrid(2e7))\r\n\r\n%%\r\nassert(squareOnGrid(2^randi(20)))\r\n\r\n%%\r\nassert(~squareOnGrid(7^(2*randi(10)+1)))\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":13,"test_suite_updated_at":"2020-12-25T06:14:57.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-24T15:45:57.000Z","updated_at":"2026-07-20T10:51:05.000Z","published_at":"2020-12-24T16:01:03.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose you have a regular grid of dots, each 1 unit apart. By connecting four dots arranged in a square of side 1, you can draw a square of area 1. Similarly, you can draw squares of area 4, 9, 16, 25, etc. As shown below, by connecting dots with diagonal lines, you can draw squares of area 2, 5, 10, 8, 13, and 17. Which squares can you draw by connecting dots on the grid with straight lines? \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes an area and determines whether the square of that area can be drawn.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.gif\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.gif\",\"contentType\":\"image/gif\",\"content\":\"https://www.mathsisfun.com/activity/images/squares8.gif\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60965,"title":"List the near-square primes, p = n² + 1 (Landau's 4th problem)","description":"Historical context\r\n\r\nAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \"unattackable at the present state of mathematics\" and are now known as Landau's problems (source : Wiki)\r\n\r\nProblem statement\r\n\r\nThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form . Your task here is to list such primes below a given input integer m.\r\n\r\nExamples\r\n\r\nm = 20 =\u003e p = [2, 5, 17] ;\r\nm = 200 =\u003e p = [2, 5, 17, 37, 101, 197] ;\r\nm = 2000 =\u003e p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\r\n\r\nFobidden functions\r\nregexp\r\nstr2num\r\nassignin\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 719.033px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 359.517px; transform-origin: 408px 359.517px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.375px 8px; transform-origin: 375.375px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \"unattackable at the present state of mathematics\" and are now known as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 65.825px 8px; transform-origin: 65.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eLandau's problems \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.725px 8px; transform-origin: 44.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(source : Wiki)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376.275px 8px; transform-origin: 376.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"110.5\" height=\"19\" style=\"width: 110.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.775px 8px; transform-origin: 211.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eYour task here is to list such primes below a given input integer\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 78.575px 8px; transform-origin: 78.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 20 =\u0026gt; p = [2, 5, 17] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 125.25px 8px; transform-origin: 125.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 200 =\u0026gt; p = [2, 5, 17, 37, 101, 197] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 230.275px 8px; transform-origin: 230.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 2000 =\u0026gt; p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e I\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = list_the_near_square_primes(m)\r\n  p = m;\r\nend","test_suite":"%%\r\nm = 20;\r\np_correct = [2,5,17];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%%\r\nm = 200;\r\np_correct = [2,5,17,37,101,197];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%%\r\nm = 2000;\r\np_correct = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('list_the_near_square_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":2,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:04:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":63,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-17T04:36:44.000Z","updated_at":"2026-07-20T06:36:47.000Z","published_at":"2025-07-17T05:04:11.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \\\"unattackable at the present state of mathematics\\\" and are now known as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eLandau's problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(source : Wiki)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = n^2 + 1, n \\\\in \\\\mathbb{N}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eYour task here is to list such primes below a given input integer\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 20 =\u0026gt; p = [2, 5, 17] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 200 =\u0026gt; p = [2, 5, 17, 37, 101, 197] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 2000 =\u0026gt; p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":50182,"title":"Compute the Euler totient function","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 123px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.5px; transform-origin: 407px 61.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.7333px 7.91667px; transform-origin: 79.7333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Euler totient function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(n)\" style=\"width: 31.5px; height: 19px;\" width=\"31.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.25px 7.91667px; transform-origin: 132.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e gives the number of integers smaller than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 7.91667px; transform-origin: 84.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that are relatively prime to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.3667px 7.91667px; transform-origin: 61.3667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e---that is, that share no common factors with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.95px 7.91667px; transform-origin: 85.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e other than 1. For example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(12) = 4\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 175.283px 7.91667px; transform-origin: 175.283px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because 1, 5, 7, and 11 share no common factors other than 1. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.0083px 7.91667px; transform-origin: 77.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a prime number, then \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(p) = p-1\" style=\"width: 85px; height: 19px;\" width=\"85\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.533px 7.91667px; transform-origin: 108.533px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because all numbers smaller than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.7917px 7.91667px; transform-origin: 70.7917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are relatively prime to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 166.333px 7.91667px; transform-origin: 166.333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the Euler totient function. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNote: I now realize that \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 656\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.517px 7.91667px; transform-origin: 174.517px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e also deals with the totient function. Apologies to Freddy.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function phi = totient(n)\r\n   phi = f(n)\r\nend","test_suite":"%%\r\nn = 1;\r\nphi_correct = 1;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 2;\r\nphi_correct = 1;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 3;\r\nphi_correct = 2;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 4;\r\nphi_correct = 2;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 6;\r\nphi_correct = 2;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 12;\r\nphi_correct = 4;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 17;\r\nphi_correct = 16;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 44;\r\nphi_correct = 20;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 100;\r\nphi_correct = 40;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 256;\r\nphi_correct = 128;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 729;\r\nphi_correct = 486;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 2310;\r\nphi_correct = 480;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 4999;\r\nphi_correct = 4998;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 12345;\r\nphi_correct = 6576;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 654321;\r\nphi_correct = 436212;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 7566179;\r\nphi_correct = 7451136;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 901800900;\r\nphi_correct = 172972800;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nn = 7956061979;\r\nphi_correct = 7944111936;\r\nassert(isequal(totient(n),phi_correct))\r\n\r\n%%\r\nassert(isequal(totient(totient(6503)),1792))\r\n\r\n%%\r\np = primes(2000);\r\nk = randi(length(p));\r\nassert(isequal(totient(p(k)),p(k)-1))\r\n\r\n%%\r\nP = primes(2000);\r\np = P(randi(length(P)));\r\nq = P(randi(length(P)));\r\nassert(isequal(totient(p*q),(p-1)*(q-1)))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-02-06T18:48:05.000Z","updated_at":"2026-07-14T12:34:15.000Z","published_at":"2021-02-06T19:13:15.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Euler totient function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e gives the number of integers smaller than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that are relatively prime to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e---that is, that share no common factors with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e other than 1. For example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(12) = 4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(12) = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because 1, 5, 7, and 11 share no common factors other than 1. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a prime number, then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(p) = p-1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(p) = p-1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because all numbers smaller than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are relatively prime to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the Euler totient function. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNote: I now realize that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 656\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e also deals with the totient function. Apologies to Freddy.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"errors":[],"facets":[[{"value":"Prime Numbers I","count":11,"selected":false},{"value":"Easy Sequences Volume IV","count":10,"selected":false},{"value":"Prime numbers properties II","count":10,"selected":false},{"value":"Easy Sequences Volume V:  Fibonacci Obsession","count":9,"selected":false},{"value":"Easy Sequences Volume VII","count":9,"selected":false},{"value":"Number theory","count":9,"selected":false},{"value":"Prime numbers properties I","count":9,"selected":false},{"value":"Easy Sequences Volume VIII","count":8,"selected":false},{"value":"Prime Numbers III","count":8,"selected":false},{"value":"Easy Sequences Volume III","count":7,"selected":false},{"value":"Easy Sequences Volume I","count":6,"selected":false},{"value":"Magic Numbers","count":6,"selected":false},{"value":"Sequences and Series VI","count":6,"selected":false},{"value":"Easy Sequences Volume II","count":5,"selected":false},{"value":"Sequences \u0026 Series V","count":5,"selected":false},{"value":"YouTube-inspired","count":5,"selected":false},{"value":"Easy Sequences Volume VI","count":4,"selected":false},{"value":"Magic Numbers IV","count":4,"selected":false},{"value":"Special Functions","count":4,"selected":false},{"value":"Magic Numbers III","count":3,"selected":false},{"value":"Algorithm I","count":2,"selected":false},{"value":"Find the nth number ","count":2,"selected":false},{"value":"Number Manipulation II","count":2,"selected":false},{"value":"Number Manipulation III","count":2,"selected":false},{"value":"Operations","count":2,"selected":false},{"value":"Prime Numbers II","count":2,"selected":false},{"value":"The Prime Directive","count":2,"selected":false},{"value":"Basics - Factorization","count":1,"selected":false},{"value":"Cody Challenge","count":1,"selected":false},{"value":"Cody5:Easy","count":1,"selected":false},{"value":"Combinatorics III","count":1,"selected":false},{"value":"Functions I","count":1,"selected":false},{"value":"Functions II","count":1,"selected":false},{"value":"High School Challenge","count":1,"selected":false},{"value":"Indexing II","count":1,"selected":false},{"value":"MATLAB Onramp Practice","count":1,"selected":false},{"value":"Number Manipulation IV","count":1,"selected":false},{"value":"Polynomials","count":1,"selected":false},{"value":"Programky -old","count":1,"selected":false},{"value":"Sequences \u0026 Series IV","count":1,"selected":false},{"value":"Special Functions II","count":1,"selected":false},{"value":"Tough Stuff","count":1,"selected":false}],[{"value":"medium","count":164,"selected":false},{"value":"hard","count":55,"selected":false},{"value":"easy","count":19,"selected":false}]],"term":"tag:\"number theory\"","page":1,"per_page":50,"sort":"map(difficulty_value,0,0,999) asc"}}