Primitive prime divisors and the $n$-th cyclotomic polynomial

Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called ${\mathrm{\Phi }}_n^{\ast }(q)$, which is closely related to the cyclotomic polynomial ${\mathrm{\Phi }}_n(x)$ and to primitive prime divisors of $q^n-1$. Our definition of ${\mathrm{\Phi }}_n^{\ast }(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we give an algorithm for determining all pairs $(n,q)$ with ${\mathrm{\Phi }}_n^{\ast }(q)\le c{n}^k$. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgroups of finite linear groups.