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.
Several papers have given empirical support to the conjecture, but it disagrees with the widely accepted (but also open) Lenstra–Pomerance–Wagstaff conjecture.
The conjecture Let be a prime number and the corresponding Mersenne prime. If , as and , the number of prime divisors of in the interval is Poisson distributed with expected value
The conjecture implies that The number of Mersenne primes less than is asymptotically . The expected number of Mersenne primes with is . The probability that is prime is .
Incompatibility with Lenstra–Pomerance–Wagstaff conjecture
The Lenstra–Pomerance–Wagstaff conjecture gives different values:Chris Caldwell, Heuristics: Deriving the Wagstaff Mersenne Conjecture. Retrieved on 2017-07-26. The number of Mersenne primes less than is asymptotically . The expected number of Mersenne primes with is . The probability that is prime is , where if and otherwise.
Asymptotically, these values are about 11% smaller.
Results While Gillie's conjecture remains open, several papers have added empirical support to its validity.
References