  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 $q$.
