Primitive root modulo n

Primitive roots modulo n are a concept from modular arithmetic, in number theory.

If n is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation; it is written as (Z/nZ)^× or Z_n^*. This group is cyclic if and only if n is equal to 2 or 4 or p^k or 2 p^k for an odd prime number p and k ≥ 1. A generator of this cyclic group, that is, all powers of some number g in Z_n^* is a number in Z_n^* is called a primitive root modulo n, or a primitive element of Z_n^*.

Take for example n = 14. The elements of (Z/14Z)^× are the congruence classes of 1, 3, 5, 9, 11 and 13. Then 3 is a primitive root modulo 14, as we have 3² = 9, 3³ = 13, 3⁴ = 11, 3⁵ = 5 and 3⁶ = 1 (modulo 14). The only other primitive root modulo 14 is 5.

Here is a table containing the smallest primitive root for various values of n (see A046145):

n	primitive root mod n
2	1
3	2
4	3
5	2
6	5
7	3
8	-
9	2
10	3
11	2
12	-
13	2
14	3

No simple general formula to compute primitive roots modulo n is known. There are however methods to locate a primitive root that are faster than simply trying out all candidates: first compute φ(n). Then determine the different prime factors of φ(n), say p₁,...,p_k. Now, for every element m of (Z/nZ)^×, compute

using the fast exponentiating by squaring. As soon as you find a number m for which these k results are all different from 1, you stop: m is a primitive root.

If the multiplicative order of a number k modulo p is p-1, then it is a primitive root. We can use this to test for primitive roots.

The number of primitive roots modulo n is equal to φ(φ(n)) since, in general, a cyclic group with r elements has φ(r) generators.

There exist positive constants C, ε and p₀ such that, for every prime p ≥ p₀, there exists a primitive root modulo p that is less than C p^1/4+ε. If the generalized Riemann hypothesis is true, then for every prime number p, there exists a primitive root modulo p that is less than 70 (ln(p))².