Gillies' conjecture

In number theory, Gillies' conjecture is a conjecture about the distribution of prime factors of Mersenne numbers. It was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good and Daniel Shanks.