Representations using Sophie Germain primes and twin prime pairs

Conjecture (i), first part: 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.

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Cite

@misc{cairn-oeis-227923,
  title        = {Representations using Sophie Germain primes and twin prime pairs},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-227923}},
  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 (i), first part: $a (n) > 0$ for all $n > 1$ .

conjecture_2. Conjecture (i), second part: Any integer $n > 4$ not equal to $13$ can be written as $x + y$ with $x$ and $y$ distinct and greater than $1$ such that $6 x - 1$ is a Sophie Germain prime and ${6 y - 1, 6 y + 1}$ is a twin prime pair.

conjecture_3. Conjecture (ii), cousin prime variant: Any integer $n > 1$ can be written as $x + y$ ( $x, y > 0$ ) such that $6 x - 1$ is a Sophie Germain prime and ${6 y + 1, 6 y + 5}$ is a cousin prime pair.

conjecture_4. Conjecture (ii), sexy prime variant: Any integer $n > 1$ can be written as $x + y$ ( $x, y > 0$ ) such that $6 x - 1$ is a Sophie Germain prime and ${6 y - 1, 6 y + 5}$ is a sexy prime pair.

Number of ways to write $n = x + y$ ( $x, y > 0$ ) such that $6 x - 1$ is a Sophie Germain prime and ${6 y - 1, 6 y + 1}$ is a twin prime pair.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«227923» (4 statements).

theorem conjecture_1 (n : ℕ) (hn : n > 1) : a n > 0

theorem conjecture_2 (n : ℕ) (hn : n > 4) (h13 : n ≠ 13) :
    ∃ x y : ℕ, 1 < x ∧ 1 < y ∧ x ≠ y ∧ x + y = n ∧
      (6 * x - 1).Prime ∧ (12 * x - 1).Prime ∧
      (6 * y - 1).Prime ∧ (6 * y + 1).Prime

theorem conjecture_3 (n : ℕ) (hn : n > 1) :
    ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ x + y = n ∧
      (6 * x - 1).Prime ∧ (12 * x - 1).Prime ∧
      (6 * y + 1).Prime ∧ (6 * y + 5).Prime

theorem conjecture_4 (n : ℕ) (hn : n > 1) :
    ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ x + y = n ∧
      (6 * x - 1).Prime ∧ (12 * x - 1).Prime ∧
      (6 * y - 1).Prime ∧ (6 * y + 5).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

A227923

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.