Generation of second maximal subgroups and the existence of special primes

Let be a finite almost simple group. It is well known that can be generated by three elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of . In this paper, we consider subgroups at the next level of the subgroup lattice - the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes for which there is a prime power such that is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.