Primitive prime divisors and the nTH cyclotomic polynomial

Abstract

Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called φ∗n(q), which is closely related to the cyclotomic polynomial φn (x) and to primitive prime divisors of qn - 1. Our definition of φ∗n(q) is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants c and k, we provide an algorithm for determining all pairs (n; q) with φ∗n(q) ≤ cnk. This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.