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.
The conjecture
- [math]\displaystyle{ \text{If } }[/math][math]\displaystyle{ A \lt B \lt \sqrt{M_p}\text{, as }B/A\text{ and }M_p \rightarrow \infty\text{, the number of prime divisors of }M }[/math]
- [math]\displaystyle{ \text{ in the interval }[A, B]\text{ is Poisson-distributed with} }[/math]
- [math]\displaystyle{ \text{mean }\sim \begin{cases} \log(\log B /\log A) & \text{ if }A \ge 2p\\ \log(\log B/\log 2p) & \text{ if } A \lt 2p \end{cases} }[/math]
He noted that his conjecture would imply that
- The number of Mersenne primes less than [math]\displaystyle{ x }[/math] is [math]\displaystyle{ ~\frac{2}{\log 2} \log\log x }[/math].
- The expected number of Mersenne primes [math]\displaystyle{ M_p }[/math] with [math]\displaystyle{ x \le p \le 2x }[/math] is [math]\displaystyle{ \sim2 }[/math].
- The probability that [math]\displaystyle{ M_p }[/math] is prime is [math]\displaystyle{ ~\frac{2 \log 2p }{p\log 2} }[/math].
Incompatibility with Lenstra–Pomerance–Wagstaff conjecture
The Lenstra–Pomerance–Wagstaff conjecture gives different values:[4][5]
- The number of Mersenne primes less than [math]\displaystyle{ x }[/math] is [math]\displaystyle{ ~\frac{e^\gamma}{\log 2} \log\log x }[/math].
- The expected number of Mersenne primes [math]\displaystyle{ M_p }[/math] with [math]\displaystyle{ x \le p \le 2x }[/math] is [math]\displaystyle{ \sim e^\gamma }[/math].
- The probability that [math]\displaystyle{ M_p }[/math] is prime is [math]\displaystyle{ ~\frac{e^\gamma\log ap}{p\log 2} }[/math] 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
- ↑ 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.
- ↑ I. J. Good (1955). "Conjectures concerning the Mersenne numbers". Mathematics of Computation 9 (51): 120–121. doi:10.1090/S0025-5718-1955-0071444-6.
- ↑ Shanks, Daniel (1962). Solved and Unsolved Problems in Number Theory. Washington: Spartan Books. pp. 198.
- ↑ Samuel S. Wagstaff (1983). "Divisors of Mersenne numbers". Mathematics of Computation 40 (161): 385–397. doi:10.1090/S0025-5718-1983-0679454-X.
- ↑ Chris Caldwell, Heuristics: Deriving the Wagstaff Mersenne Conjecture. Retrieved on 2017-07-26.
- ↑ 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.
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