Primitive prime divisors and the nTH cyclotomic polynomial

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 φ _n (x) and to primitive prime divisors of q ⁿ - 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) ≤ cn ^k. 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.