Misplaced Pages

Gillies' conjecture

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
This article needs attention from an expert in Mathematics. The specific problem is: Needs to be checked by editor with advanced mathematics knowledge. WikiProject Mathematics may be able to help recruit an expert. (January 2014)

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

If  {\displaystyle {\text{If }}} A < B < M p , as  B / A  and  M p → ∞ , the number of prime divisors of  M {\displaystyle A<B<{\sqrt {M_{p}}}{\text{, as }}B/A{\text{ and }}M_{p}\rightarrow \infty {\text{, the number of prime divisors of }}M}
 in the interval  [ A , B ]  is Poisson-distributed with {\displaystyle {\text{ in the interval }}{\text{ is Poisson-distributed with}}}
mean  ∼ { log ⁡ ( log ⁡ B / log ⁡ A )  if  A ≥ 2 p log ⁡ ( log ⁡ B / log ⁡ 2 p )  if  A < 2 p {\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 x {\displaystyle x} is   2 log ⁡ 2 log ⁡ log ⁡ x {\displaystyle ~{\frac {2}{\log 2}}\log \log x} .
  2. The expected number of Mersenne primes M p {\displaystyle M_{p}} with x ≤ p ≤ 2 x {\displaystyle x\leq p\leq 2x} is ∼ 2 {\displaystyle \sim 2} .
  3. The probability that M p {\displaystyle M_{p}} is prime is   2 log ⁡ 2 p p log ⁡ 2 {\displaystyle ~{\frac {2\log 2p}{p\log 2}}} .

Incompatibility with Lenstra–Pomerance–Wagstaff conjecture

The Lenstra–Pomerance–Wagstaff conjecture gives different values:

  1. The number of Mersenne primes less than x {\displaystyle x} is   e γ log ⁡ 2 log ⁡ log ⁡ x {\displaystyle ~{\frac {e^{\gamma }}{\log 2}}\log \log x} .
  2. The expected number of Mersenne primes M p {\displaystyle M_{p}} with x ≤ p ≤ 2 x {\displaystyle x\leq p\leq 2x} is ∼ e γ {\displaystyle \sim e^{\gamma }} .
  3. The probability that M p {\displaystyle M_{p}} is prime is   e γ log ⁡ a p p log ⁡ 2 {\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.

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.
Categories: