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 (but also open) 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.

References