  We use the Bateman--Horn Conjecture from number theory to give strong
evidence of a positive answer to Peter Neumann's question, whether there are
infinitely many simple groups of order a product of six primes. (Those with
fewer than six were classified by Burnside, Frobenius and H\"older in the
1890s.) The groups satisfying this condition are ${\rm PSL}_2(8)$, ${\rm
PSL}_2(9)$ and ${\rm PSL}_2(p)$ for primes $p$ such that $p^2-1$ has just six
prime factors. The conjecture suggests that there are infinitely many such
primes, by providing heuristic estimates for their distribution which agree
closely with evidence from computer searches. We also briefly discuss the
applications of this conjecture to other problems in group theory, such as the
classifications of permutation groups and of linear groups of prime degree, the
structure of the power graph of a finite simple group, and the construction of
highly symmetric block designs.
