Lemoine's conjectures — open problem in number theory

Lemoine's conjectures

For all odd integers n ≥ 7 there are prime numbers p,q such that n = p+2q.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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-lemoine,
  title        = {Lemoine's conjectures},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/lemoine}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

lemoine_conjecture. For all odd integers $n \geq 7$ there are prime numbers $p, q$ such that $n = p + 2 q$ .

lemoine_conjecture_extension. For all odd integers $n \geq 9$ there are odd prime numbers $p, q, r, s$ and natural numbers $a, b$ such that $p + 2 q = n$ , $2 + pq = 2^{a} + r$ , $2 p + q = 2^{b} + s$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Lemoine (2 statements).

theorem lemoine_conjecture (n : ℕ) (hn : 6 < n) (odd : Odd n) :
    ∃ (p q : ℕ), p.Prime ∧ q.Prime ∧ p + 2 * q = n

theorem lemoine_conjecture_extension (n : ℕ) (hn : 8 < n) (odd : Odd n) :
    ∃ (p q r s a b : ℕ), OddPrime p ∧ OddPrime q ∧ OddPrime r ∧ OddPrime s ∧
    p + 2 * q = n ∧ 2 + p * q = 2 ^ a + r ∧ 2 * p + q = 2 ^ b + s

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.

References

Wikipedia
[Ki85] Kiltinen, J. and Young P. (1985). Goldbach, Lemoine, and a Know/Don't Know Problem.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.