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
Famous quotes containing the word definition:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)