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

 * $$\text{If }$$$$A < B < \sqrt{M_p}\text{, as }B/A\text{ and }M_p \rightarrow \infty\text{, the number of prime divisors of }M$$
 * $$\text{ in the interval }[A, B]\text{ is Poisson-distributed with}$$

\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 $$~\frac{2 \log 2p }{p\log 2}$$.

Incompatibility with Lenstra–Pomerance–Wagstaff conjecture
The Lenstra–Pomerance–Wagstaff conjecture gives different values:
 * 1) The number of Mersenne primes less than $$x$$ is $$~\frac{e^\gamma}{\log 2} \log\log x$$.
 * 2) The expected number of Mersenne primes $$M_p$$ with $$x \le p \le 2x$$ is $$\sim e^\gamma$$.
 * 3) The probability that $$M_p$$ is prime is $$~\frac{e^\gamma\log ap}{p\log 2}$$ 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.