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

He noted that his conjecture would imply that

  1. The number of Mersenne primes less than is .
  2. The expected number of Mersenne primes with is .
  3. The probability that is prime is .

Incompatibility with Lenstra–Pomerance–Wagstaff conjecture

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

  1. The number of Mersenne primes less than is .
  2. The expected number of Mersenne primes with is .
  3. 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

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.
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