Ways to write n = k + m with p_k^2 - 2, p_m^2 - 2, and p_p_m^2

Ways to write n = k + m with p_k^2 - 2, p_m^2 - 2, and p_p_m^2 - 2 all prime

Conjecture: a(n) > 0 for all n > 1.

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-237413,
  title        = {Ways to write n = k + m with p_k^2 - 2, p_m^2 - 2, and p_p_m^2 - 2 all prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-237413}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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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

Conjecture: $a (n) > 0$ for all $n > 1$ .

Let $p_{j}$ denote the $j$ -th prime number. The sequence $a (n)$ counts the number of ways to write $n = k + m$ with $k > 0$ and $m > 0$ such that $p_{k}^{2} - 2$ , $p_{m}^{2} - 2$ , and $p_{p_{m}}^{2} - 2$ are all prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«237413».

theorem conjecture (n : ℕ) (hn : 1 < n) : 0 < a n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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

A237413

Source and licence

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