Primes in geometric series and finite permutation groups

TL;DR

This paper explores the occurrence of primes in specific geometric series related to finite permutation groups, proposing a conjecture supported by heuristics and computational evidence about their infinitude.

Contribution

It introduces a conjecture that infinitely many primes of a certain form related to finite simple groups exist, supported by heuristic and computational evidence.

Findings

Heuristic arguments suggest the conjecture is plausible.

Computational evidence supports the existence of infinitely many such primes.

The problem remains open and is connected to the classification of permutation groups.

Abstract

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm L}_n(q)$ is prime. We present heuristic arguments and computational evidence to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$.