compute a primitive root - Maple Help

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numtheory[primroot] - compute a primitive root

 Calling Sequence primroot(g, n) primroot(n)

Parameters

 g - positive integer or 0 n - integer greater than 1

Description

 • The function primroot will compute the first primitive root of n that is greater than g, if possible, otherwise it returns FAIL.  The integers that are relatively prime to n form a group of order $\mathrm{\phi }\left(n\right)$ under multiplication mod n. If this group is cyclic then a generator of the group is called a primitive root of n (i.e. the order of primroot (g, n) is $\mathrm{\phi }\left(n\right)$).  If only one argument n is present (in this case $g=0$) then this function will return the smallest primitive root of the number n.
 • The command with(numtheory,primroot) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{numtheory}\right):$
 > $\mathrm{primroot}\left(2\right)$
 ${1}$ (1)
 > $\mathrm{primroot}\left(41\right)$
 ${6}$ (2)
 > $\mathrm{primroot}\left(0,41\right)$
 ${6}$ (3)
 > $\mathrm{primroot}\left(7,41\right)$
 ${11}$ (4)
 > $\mathrm{order}\left(,41\right)$
 ${40}$ (5)
 > $\mathrm{φ}\left(41\right)$
 ${40}$ (6)
 > $\mathrm{primroot}\left(2,8\right)$
 ${\mathrm{FAIL}}$ (7)