Gillies' conjecture


In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and 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. The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted Lenstra–Pomerance–Wagstaff conjecture.

The conjecture

He noted that his conjecture would imply that
  1. The number of Mersenne primes less than is.
  2. The expected number of Mersenne primes with is.
  3. The probability that is prime is.

Incompatibility with Lenstra–Pomerance–Wagstaff conjecture

The Lenstra–Pomerance–Wagstaff conjecture gives different values:
  1. The number of Mersenne primes less than is.
  2. The expected number of Mersenne primes with is.
  3. The probability that is prime is with a = 2 if p = 3 mod 4 and 6 otherwise.
Asymptotically these values are about 11% smaller.

Results

While Gillie's conjecture remains open, several papers have added empirical support to its validity, including Ehrman's 1964 paper.