Primitive prime divisors and the $n$-th 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 $Φ_n(x)$ 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 give an algorithm for determining all pairs $(n,q)$ with $Φ^*_n(q)\le cn^k$. 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.

14 pages, 5 tables in Journal of the Australian Mathematical Society We replaced $n>1$ with $n>2$ in the statement of Bang's theorem on page 2 (we thank Tim Penttila for pointing this out)