The PseudoPrimitiveRoot(n) command returns the smallest pseudo primitive root modulo n, if it exists.

The PseudoPrimitiveRoot(n, greaterthan = m) command returns the smallest pseudo primitive root modulo n greater than m.

The PseudoPrimitiveRoot(n, ith = i) command returns the ith smallest pseudo primitive root modulo n.

If the required pseudo primitive root does not exist, then an error message is displayed.

A pseudo primitive root is an integer $y$ such that $y$ and $n$ are coprime, and there does not exist an integer $x$ such that $x^r=y\mathbf{mod}n$ where $r$ is a divisor of $n$ not equal to $1$.

If a primitive root modulo n exists, then the pseudo primitive roots are exactly the primitive roots.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{PseudoPrimitiveRoot}\left(7\right)$

$3$

$\mathrm{PseudoPrimitiveRoot}\left(2662\,\mathrm{greaterthan}=2342\right)$

$2345$

$\mathrm{PrimitiveRoot}\left(8\right)$

Error, (in NumberTheory:-PrimitiveRoot) there does not exist a primitive root modulo 8

$\mathrm{seq}\left(\mathrm{PseudoPrimitiveRoot}\left(8\,\mathrm{ith}=i\right)\,i=1..3\right)$

$3,5,7$

$\mathrm{PseudoPrimitiveRoot}\left(8\,\mathrm{ith}=4\right)$

Error, (in NumberTheory:-PseudoPrimitiveRoot) there exist only 3 pseudo primitive roots modulo 8

$\mathrm{PseudoPrimitiveRoot}\left(8\,\mathrm{greaterthan}=7\right)$

Error, (in NumberTheory:-PseudoPrimitiveRoot) there does not exist a pseudo primitive root modulo 8 greater than 7

The NumberTheory[PseudoPrimitiveRoot] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.