Conjectures about Mersenne primes

For any odd natural number p if two of the following conditions hold, then all three must hold: 1. 2^p-1 is prime 2. (2^p+1)/3 is prime 3. Exists a number k such that p = 2^k pm 1 or p = 4^k pm 3

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-mersenne,
  title        = {Conjectures about Mersenne primes},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/mersenne}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

new_mersenne_conjecture. For any odd natural number p if two of the following conditions hold, then all three must hold:

$2^{p} - 1$ is prime
$(2^{p} + 1) /3$ is prime
Exists a number k such that $p = 2^{k} p m 1$ or $p = 4^{k} p m 3$

infinitely_many_mersenne_primes. Are there infinitely many Mersenne primes?

catalans_mersenne_conjecture. The first five Catalan-Mersenne numbers $c_{0}, \dots, c_{4}$ are known to be prime. Catalan conjectured that they are prime "up to a certain limit". Are all Catalan-Mersenne numbers $c_{n}$ with $n \geq 5$ prime?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Mersenne (3 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem new_mersenne_conjecture (p : ℕ) (hp : Odd p) :
    NewMersenneConjectureStatement p

theorem infinitely_many_mersenne_primes :
  answer(sorry) ↔ Set.Infinite { p : ℕ | (mersenne p).Prime }

theorem catalans_mersenne_conjecture :
    answer(sorry) ↔ ∀ n ≥ 5, Nat.Prime (catalanMersenne n)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

new_mersenne_conjecture.variants.prime — The New Mersenne Conjecture statement holds for odd primes.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.