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$$A < B < \sqrt\textB/A\textM_p \rightarrow \infty\textM$
:$\text$, Btext
::$\text\sim \begin \log(\log B /\log A) & \textA \ge 2p\\ \log(\log B/\log 2p) & \text A < 2p \end$

He noted that his conjecture would imply that
# The number of Mersenne primes less than $x$ is $~\frac \log\log x$.
# The expected number of Mersenne primes $M_p$ with $x \le p \le 2x$ is $\sim2$.
# The probability that $M_p$ is prime is $~\frac$.

Incompatibility with Lenstra–Pomerance–Wagstaff conjecture

The Lenstra–Pomerance–Wagstaff conjecture gives different values:Chris Caldwell
Heuristics: Deriving the Wagstaff Mersenne Conjecture
Retrieved on 2017-07-26.
# The number of Mersenne primes less than $x$ is $~\frac \log\log x$.
# The expected number of Mersenne primes $M_p$ with $x \le p \le 2x$ is $\sim e^\gamma$.
# The probability that $M_p$ is prime is $~\frac$ 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.

References

{{DEFAULTSORT:Gillies Conjecture
Conjectures
Unsolved problems in number theory
Hypotheses
Mersenne primes