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Primitive Root Modulo NA primitive root modulo ''n'' is a concept from modular arithmetic in number theory. If n≥1 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 Zn*. This group is cyclic if and only if n is equal to 1 or 2 or 4 or pk or 2 pk for an odd prime number p and k ≥ 1. A generator of this cyclic group is called a primitive root modulo n, or a primitive element of Zn*. A primitive root modulo n, in other words, is an integer g such that, modulo n, every integer not having a common factor with n is congruent to a power of g. 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> | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | | primitive root mod n | 1 | 2 | 3 | 2 | 5 | 3 | - | 2 | 3 | 2 | - | 2 | 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. If the multiplicative order of a number m modulo n is equal to φ(n) (the order of Z/nZ), then it is a primitive root. We can use this to test for primitive roots: - first compute φ(n). Then determine the different prime factors of φ(n), say p1,...,pk. 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. The number of primitive roots modulo n, if there are any, is equal to - φ(φ(n))
since, in general, a cyclic group with r elements has φ(r) generators. Sometimes one is interested in small primitive roots. We have the following results. For every ε>0 there exist positive constants C and p0 such that, for every prime p ≥ p0, there exists a primitive root modulo p that is less than - C p1/4+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.
See also: Artin conjecture.
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