gives the smallest primitive root of n greater than or equal to k.
- PrimitiveRoot[n] gives a generator for the multiplicative group of integers modulo n relatively prime to n.
- PrimitiveRoot[n] returns unevaluated if n is not 2, 4, an odd prime power, or twice an odd prime power.
- PrimitiveRoot[n,1] computes the smallest primitive root of n.
Examplesopen allclose all
Basic Examples (2)
A primitive root of 9 is 2:
The primitive root generates all integers modulo 9 that are relatively prime to 9:
A primitive root of 10:
The smallest primitive root of 10:
Find the smallest primitive root:
Find the primitive root greater than a number:
PrimitiveRoot works on large integers:
PrimitiveRoot automatically threads over lists:
Properties & Relations (2)
The multiplicative order of a primitive root modulo n is EulerPhi[n]:
For a prime p, there exist EulerPhi[p-1] primitive roots modulo p:
Neat Examples (1)
Elements relatively prime to 22 are enumerated by the primitive root:
Introduced in 2007
Updated in 2015