# isPrimitiveRoot

Determine which array elements are primitive roots

## Syntax

``TF = isPrimitiveRoot(G,N)``

## Description

example

````TF = isPrimitiveRoot(G,N)` returns a logical array containing `1 (true)` for the corresponding elements of `G` that are primitive roots modulo `N`, and `0 (false)` for the corresponding elements that are not primitive roots. The elements of `G` must be integers, and the elements of `N` must be positive integers.```

## Examples

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Create a row vector containing positive integers from 1 to 11. Determine if they are primitive roots modulo 11.

```G = 1:11; TF = isPrimitiveRoot(G,11)```
```TF = 1x11 logical array 0 1 0 0 0 1 1 1 0 0 0 ```

Find the smallest positive integer that is a primitive root modulo 11.

`Z1 = find(TF,1)`
```Z1 = 2 ```

Show all positive integers (less than or equal to 11) that are primitive roots modulo 11.

`Z = G(TF)`
```Z = 1×4 2 6 7 8 ```

Create a row vector containing integers from –15 to 15. Find the integers that are primitive roots modulo 15.

```G = -15:15; Z = G(isPrimitiveRoot(G,15))```
```Z = 1x0 empty double row vector ```

The integer 15 has no primitive roots.

## Input Arguments

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Base, specified as a number, vector, matrix, array, symbolic number, or symbolic array. The elements of `G` must be integers. `G` and `N` must be the same size, or else one of them must be a scalar.

Data Types: `single` | `double` | `sym`

Divisor, specified as a number, vector, matrix, array, symbolic number, or symbolic array. The elements of `N` must be positive integers. `G` and `N` must be the same size, or else one of them must be a scalar.

Data Types: `single` | `double` | `sym`

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### Primitive Root

A number g is a primitive root modulo n if every number a coprime to n (or $\mathrm{gcd}\left(a,n\right)=1$) is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is an integer k such that . The primitive roots modulo n exist if and only if , where p is an odd prime and k is a positive integer.

For example, the integer 2 is a primitive root modulo 5 because is satisfied for every integer a that is coprime to 5.