The problem
Question: are there infinitely many primes of the form \(2^{p}\) − 1? The Lenstra–Pomerance–Wagstaff conjecture refines it: infinitely many, with about e^γ log₂ log₂ x of them below x, and correspondingly structured exponents.
Are There Infinitely Many Mersenne Primes?
openHistory & significance
Mersenne's 1644 list launched the subject; Lucas (1876) and Lehmer turned testing into theory; every largest-known-prime record since 1952 has been a Mersenne prime, found by increasingly distributed computation — culminating in October 2024 when GIMPS volunteer Luke Durant, running the search on GPU cloud infrastructure, found the 52nd: \(2^{136279841}\) − 1, over 41 million digits. Heuristics strongly assert inexhaustibility; a single honest proof remains as remote as ever.
Still open.
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