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arXiv · 1504.02598

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

S. P. Glasby·Frank Lübeck·Alice C. Niemeyer·Cheryl E. Praeger
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Abstract

Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $\Phi^*_n(q)$, which is closely related to the cyclotomic polynomial $\Phi_n(x)$ and to primitive prime divisors of $q^n-1$. Our definition of $\Phi^*_n(q)$ 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 $\Phi^*_n(q)\le cn^k$. 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.

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S. P. Glasby, Frank Lübeck, Alice C. Niemeyer, Cheryl E. Praeger. 2015-04-10. Primitive prime divisors and the $n$-th cyclotomic polynomial. https://doi.org/10.1017/s1446788715000269

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