Gillies' conjecture explained

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^[1] 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^[2] and Daniel Shanks.^[3] 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

If A < B < \sqrt{M _{p},asB/AandM} _{p} → inf ty ,thenumberofprimedivisorsof M intheinterval [A, B] isPoisson-distributedwith mean \sim \begin{cases} log ( log B / log A) & if A \ge 2 p\\ log ( log B/ log 2 p) & if A < 2 p \end{cases}

He noted that his conjecture would imply that

1. The number of Mersenne primes less than

x is ~ (2)/(log2) loglog x .

1. The expected number of Mersenne primes

M _{p} with x \le p \le 2 x is \sim 2 .

1. The probability that

M _{p} is prime is ~ (2log2 p)/(p log2) .

Incompatibility with Lenstra–Pomerance–Wagstaff conjecture

The Lenstra–Pomerance–Wagstaff conjecture gives different values:^{[4] [5]}

1. The number of Mersenne primes less than

x is ~ (e ^{\gamma})/(log2) loglog x .

1. The expected number of Mersenne primes

M _{p} with x \le p \le 2 x is \sim e ^{\gamma} .

1. The probability that

M _{p} is prime is ~ (e ^{\gammalog} ap)/(p log2) 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.^[6]

Notes and References

1. Donald B. Gillies. Three new Mersenne primes and a statistical theory. Mathematics of Computation. 18. 93–97. 1964. 10.1090/S0025-5718-1964-0159774-6. 85. free.
2. I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation. 9. 120–121. 1955. 10.1090/S0025-5718-1955-0071444-6. 51. free.
3. Book: Shanks , Daniel . 1962. Solved and Unsolved Problems in Number Theory. Spartan Books. Washington. 198.
4. Samuel S. Wagstaff. Divisors of Mersenne numbers. Mathematics of Computation. 40. 385–397. 1983. 10.1090/S0025-5718-1983-0679454-X. 161. free.
5. Chris Caldwell, Heuristics: Deriving the Wagstaff Mersenne Conjecture. Retrieved on 2017-07-26.
6. John R. Ehrman. The number of prime divisors of certain Mersenne numbers. Mathematics of Computation. 21. 700–704. 1967. 10.1090/S0025-5718-1967-0223320-1. 100. free.