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: several papers give empirical support, but it disagrees with the widely accepted (but also open) Lenstra–Pomerance–Wagstaff conjecture.

The conjecture

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

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

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

He noted that his conjecture would imply that

1. The number of Mersenne primes less than ${\displaystyle x}$ is ${\displaystyle ~{\frac {2}{\log 2}}\log \log x}$.
2. The expected number of Mersenne primes ${\displaystyle M_{p}}$ with ${\displaystyle x\leq p\leq 2x}$ is ${\displaystyle \sim 2}$.
3. The probability that ${\displaystyle M_{p}}$ is prime is ${\displaystyle ~{\frac {2\log 2p}{p\log 2}}}$.

Incompatibility with Lenstra–Pomerance–Wagstaff conjecture

The Lenstra–Pomerance–Wagstaff conjecture gives different values:^[4][5]

1. The number of Mersenne primes less than ${\displaystyle x}$ is ${\displaystyle ~{\frac {e^{\gamma }}{\log 2}}\log \log x}$.
2. The expected number of Mersenne primes ${\displaystyle M_{p}}$ with ${\displaystyle x\leq p\leq 2x}$ is ${\displaystyle \sim e^{\gamma }}$.
3. The probability that ${\displaystyle M_{p}}$ is prime is ${\displaystyle ~{\frac {e^{\gamma }\log ap}{p\log 2}}}$ 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.^[6]

References

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. Samuel S. Wagstaff (1983). "Divisors of Mersenne numbers". Mathematics of Computation. 40 (161): 385–397. doi:10.1090/S0025-5718-1983-0679454-X.
5. Chris Caldwell, Heuristics: Deriving the Wagstaff Mersenne Conjecture. Retrieved on 2017-07-26.
6. 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.