Primes in geometric series and finite 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 ${\mathrm{L}}_n(q)$ 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 $q$.

arXiv comment25 pages, 6 figures. Version 2 contains extra material concerning primality testing, with relevant citations in the extended bibliography