Primitive root modulo n

A 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 Template:OEIS:

 n primitive root mod n 2 3 4 5 6 7 8 9 10 11 12 13 14 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
[itex]m^{\phi(n)/p_i}\mod n \qquad\mbox{ for } i=1,\ldots,k[itex]

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 pp0, 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.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy