Open problem, Arithmetic and number theory, posed 1644

Infinitude of Mersenne primes

Open

There are infinitely many Mersenne primes, primes of the form $M_p = 2^p - 1$ where $p$ itself is prime.

Research frontier as of 2026

As of 2026 the infinitude of Mersenne primes is open. Only 52 are known, found by testing every exponent up to roughly $136$ million with the Lucas–Lehmer test through GIMPS. The Lenstra–Pomerance–Wagstaff heuristic, treating each $M_p$ as prime with the 'expected' probability $\frac{e^\gamma}{\ln 2}\cdot\frac{1}{p}$ for prime $p$ , predicts about $\frac{e^\gamma}{\ln 2}\ln X$ Mersenne primes with exponent up to $X$ — a divergent, hence infinite, count — and matches the known list well, but it is a probabilistic model, not a proof.

Best known results

52 known Mersenne primes as of 2024, the largest being $2^{136279841}-1$ with $41{,}024{,}320$ digits (Luke Durant / GIMPS, 2024).
The Lenstra–Pomerance–Wagstaff heuristic predicts infinitely many Mersenne primes and matches the observed count well, but remains unproven.

Tools and where they stop

Tool	Achieved	Where it stops
Lucas–Lehmer primality test	Gives a fast, deterministic test for whether a specific $M_p$ is prime, letting GIMPS check every exponent up to the current search frontier.	Only ever tests finitely many candidates one at a time; it gives no way to prove that the search never runs dry.
Lenstra–Pomerance–Wagstaff heuristic	Predicts the expected count and spacing of Mersenne primes up to a bound, matching the historical record well.	It is a probabilistic model assuming independence that is not rigorously justified, so it cannot settle the infinitude question.

Open questions

Is there any proof, even conditional on a standard conjecture, that infinitely many Mersenne primes exist?
Does an odd perfect number exist, and if not, can that be shown without first resolving the Mersenne prime question?

References

Chris K. Caldwell (2024). Mersenne Primes: History, Theorems and Lists
Samuel S. Wagstaff Jr. (1983). Divisors of Mersenne numbers · DOI:10.1090/S0025-5718-1983-0679454-X
Great Internet Mersenne Prime Search (2024). GIMPS Discovers Largest Known Prime Number: 2^136279841-1