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
[edit]He noted that his conjecture would imply that
- The number of Mersenne primes less than is .
- The expected number of Mersenne primes with is .
- The probability that is prime is .
Incompatibility with Lenstra–Pomerance–Wagstaff conjecture
[edit]The Lenstra–Pomerance–Wagstaff conjecture gives different values:[4][5]
- The number of Mersenne primes less than is .
- The expected number of Mersenne primes with is .
- The probability that is prime is with a = 2 if p = 3 mod 4 and 6 otherwise.
Asymptotically these values are about 11% smaller.
Results
[edit]While Gillie's conjecture remains open, several papers have added empirical support to its validity, including Ehrman's 1964 paper.[6]
References
[edit]- ^ 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. p. 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.