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

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

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

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

The integers that are coprime to n form a group of order Totient(n) under multiplication modulo n. If this group is cyclic, then a generator is called a primitive root modulo n. That is, if p is a primitive root modulo n, then every integer coprime to n is congruent to some power of p modulo n.

If a primitive root modulo n exists, then the number of primitive roots is Totient(Totient(n)).

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

$\mathrm{PrimitiveRoot}\left(4\right)$

$3$

$\mathrm{Totient}\left(\mathrm{Totient}\left(4\right)\right)$

$1$

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

$3$

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

$2$

$\mathrm{PrimitiveRoot}\left(7\,\mathrm{greaterthan}=3\right)$

$5$

$\left[\mathrm{seq}\left(3^i\mathbf{mod}7\,i=1..\mathrm{Totient}\left(7\right)\right)\right],\left[\mathrm{seq}\left(5^i\mathbf{mod}7\,i=1..\mathrm{Totient}\left(7\right)\right)\right]$

$\left[3\,2\,6\,4\,5\,1\right],\left[5\,4\,6\,2\,3\,1\right]$

$\left[\mathrm{seq}\left(3^n\mathbf{mod}8\,n=0..2\right)\right],\left[\mathrm{seq}\left(5^n\mathbf{mod}8\,n=0..2\right)\right],\left[\mathrm{seq}\left(7^n\mathbf{mod}8\,n=0..2\right)\right]$

$\left[1\,3\,1\right],\left[1\,5\,1\right],\left[1\,7\,1\right]$

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

$4$

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

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

$\mathrm{PrimitiveRoot}\left(27\right)$

$2$

$\mathrm{seq}\left(\mathrm{PrimitiveRoot}\left(27\,\mathrm{ith}=i\right)\,i=1..\mathrm{Totient}\left(\mathrm{Totient}\left(27\right)\right)\right)$

$2,5,11,14,20,23$

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

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