SEMI-PRIMITIVE ROOT MODULO n

Abstract

Consider a multiplicative group of integers modulo n, denoted by Z n. Any element a 2 Z n is said to be a semi-primitive root if the order of a modulo n is (n)=2, where (n) is the Euler phi-function. In this paper, we classify the multiplicative groups of integers having semi-primitive roots and give interesting properties of such groups. Given a positive integer n, the integers between 1 and n which are coprime ton form a group with multiplication modulon as the operation (4); it is denoted by Z n and is called the multiplicative group of integers modulo n. For any integer a coprime to n, Euler's theorem states that a (n) 1 mod n, where (n) is the Euler phi-function (1), that is, the number of elements in Z n and a is said to be a primitive root modulo n if the order of a modulo n is equal to (n). It is well known (5) that Z n has a primitive root, equivalently, Z n is cyclic if and only if n is equal to 1, 2, 4, p k , or 2p k where p k is a power of an odd prime number. This leaves us questions about Z n that does not possess any primitive roots.