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, although several papers have added empirical support to its validity.
The conjecture
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 .
Known results
While Gillie's conjecture remains an open problem, several papers have added empirical support to its validity, including Ehrman's 1964 paper[4] as well as Wagstaff's 1983 paper.[5]
Notes
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