Neha Patil (Editor)

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 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, although several papers have added empirical support to its validity.

Contents

The conjecture

If  A < B < M p , as  B / A  and  M p → ∞ , the number of prime divisors of  M  in the interval  [ A , B ]  is Poisson-distributed with mean  ∼ { log ⁡ ( log ⁡ B / log ⁡ A )  if  A ≥ 2 p log ⁡ ( log ⁡ B / log ⁡ 2 p )  if  A < 2 p

He noted that his conjecture would imply that

  1. The number of Mersenne primes less than x is   2 log ⁡ 2 log ⁡ log ⁡ x .
  2. The expected number of Mersenne primes M p with x ≤ p ≤ 2 x is ∼ 2 .
  3. The probability that M p is prime is   2 log ⁡ 2 p p log ⁡ 2 .

Known results

While Gillie's conjecture remains an open problem, several papers have added empirical support to its validity, including Ehrman's 1964 paper as well as Wagstaff's 1983 paper.

References

Gillies' conjecture Wikipedia
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