Ordered ways to write n = k + m with p_k + 4 and p_p_m + 4 both prime

Conjecture: For each d = 1, 2, 3, … there is a positive integer N(d) for which any integer n > N(d) can be written as k + m with k > 0 and m > 0 such that p_k + 2d and p_p_m + 2d are both prime.

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-oeis-237348,
  title        = {Ordered ways to write n = k + m with p_k + 4 and p_p_m + 4 both prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-237348}},
  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_1. Conjecture: For each $d = 1, 2, 3, \dots$ there is a positive integer $N (d)$ for which any integer $n > N (d)$ can be written as $k + m$ with $k > 0$ and $m > 0$ such that $p_{k} + 2 d$ and $p_{p_{m}} + 2 d$ are both prime.

conjecture_2. In particular, we may take $(N (1), N (2), \dots, N (10)) = (2, 11, 4, 15, 31, 4, 2, 77, 4, 7)$ in the conjecture.

Let $p_{j}$ denote the $j$ -th prime number. The sequence $a (n)$ counts the number of ordered ways to write $n = k + m$ with $k > 0$ and $m > 0$ such that $p_{k} + 4$ and $p_{p_{m}} + 4$ are both prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«237348» (2 statements).

theorem conjecture_1 (d : ℕ) (hd : 1 ≤ d) :
    ∃ N : ℕ, 0 < N ∧ ∀ n : ℕ, N < n →
      ∃ k m : ℕ, 0 < k ∧ 0 < m ∧ n = k + m ∧
        (Nat.nth Nat.Prime (k - 1) + 2 * d).Prime ∧
        (Nat.nth Nat.Prime (Nat.nth Nat.Prime (m - 1) - 1) + 2 * d).Prime

theorem conjecture_2 (d : Fin 10) (n : ℕ)
    (hn : ![2, 11, 4, 15, 31, 4, 2, 77, 4, 7] d < n) :
    ∃ k m : ℕ, 0 < k ∧ 0 < m ∧ n = k + m ∧
      (Nat.nth Nat.Prime (k - 1) + 2 * (d.val + 1)).Prime ∧
      (Nat.nth Nat.Prime (Nat.nth Nat.Prime (m - 1) - 1) + 2 * (d.val + 1)).Prime

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

A237348

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.