The number of Sylow subgroups and a generalization ofMersenne primes

Fix an integer m>2. We prove that if there exists a finite group with mp+1 Sylow p-subgroups, where p is large enough, then mp+1 is prime. This improves on a theorem of M. Hall and is a partial answer to Brauer’s Problem 26. Our proof uses techniques from analytic number theory, and it also raises new questions in that area.