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\to \mathrm{\infty }\text{, the number of prime divisors of }M$

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

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

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 $\sim 2$.
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.