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^[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, although several papers have added empirical support to its validity.

The conjecture

$\text{If }$$A < B < \sqrt{M_p}\text{, as }B/A\text{ and }M_p \rightarrow \infty\text{, the number of prime divisors of }M$

$\text{ in the interval }[A, B]\text{ is Poisson-distributed with}$

$\text{mean }\sim \begin{cases} \log(\log B /\log A) & \text{ if }A \ge 2p\\ \log(\log B/\log 2p) & \text{ if } A < 2p \end{cases}$

He noted that his conjecture would imply that

1. The number of Mersenne primes less than $x$ is $~\frac{2}{\log 2} \log\log x$.
2. The expected number of Mersenne primes $M_p$ with $x \le p \le 2x$ is $\sim2$.
3. The probability that $M_p$ is prime is $~\frac{2 \log 2p }{p\log 2}$.

Known results

While Gillie's conjecture remains an open problem, several papers have added empirical support to its validity, including Ehrman's 1964 paper^[4] as well as Wagstaff's 1983 paper.^[5]

Notes

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