Definition
If n is a positive integer, the integers between 1 and n−1 which are coprime to n (or equivalently, the congruence classes coprime to n) form a group with multiplication modulo n as the operation; it is denoted by Zn× and is called the group of units modulo n or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this group is cyclic if and only if n is equal to 2, 4, pk, or 2 pk where pk is a power of an odd prime number. A generator of this cyclic group is called a primitive root modulo n, or a primitive element of Zn×.
The order of (i.e. the number of elements in) Zn× is given by Euler's totient function Euler's theorem says that aφ(n) ≡ 1 (mod n) for every a coprime to n; the lowest power of a which is congruent to 1 modulo n is called the multiplicative order of a modulo n. In particular, for a to be a primitive root modulo n, φ(n) has to be the smallest power of a which is congruent to 1 modulo n.
Read more about this topic: Primitive Root Modulo n
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