Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[MultiplicativeOrder] instead.

The smallest integer i such that $n^i=1\mathbf{mod}m$ is returned.

More formally, the order of n in the multiplicative group mod m is returned. If n and m are not coprime, then FAIL is returned.

The command with(numtheory,order) allows the use of the abbreviated form of this command.

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

$\mathrm{order}\left(13\,100\right)$

$20$

$\mathrm{order}\left(5\,8\right)$

$2$

$\mathrm{order}\left(8\,12\right)$

$\mathrm{FAIL}$

$\mathrm{primroot}\left(20\,239\right)$

$21$

$\mathrm{order}\left(\,239\right)$

$238$

$\mathrm{\phi }\left(239\right)$

$238$