Gillies' conjecture

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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.

{\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\geq 2p\\\log(\log B/\log 2p)&{\text{ if }}A<2p\end{cases}}

He noted that his conjecture would imply that

The number of Mersenne primes less than $x$ is $~{\frac {2}{\log 2}}\log \log x$ .
The expected number of Mersenne primes $M_{p}$ with $x\leq p\leq 2x$ is $\sim 2$ .
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:^[4]^[5]

The number of Mersenne primes less than $x$ is $~{\frac {e^{\gamma }}{\log 2}}\log \log x$ .
The expected number of Mersenne primes $M_{p}$ with $x\leq p\leq 2x$ is $\sim e^{\gamma }$ .
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.

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