Number of primes p < n such that p φ(n - p) - 1 is also prime

Conjecture: a(n) > 0 for all n > 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-oeis-233566,
  title        = {Number of primes p < n such that p φ(n - p) - 1 is also prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-233566}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

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Disputed

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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: $a (n) > 0$ for all $n > 3$ .

conjecture_2. Conjecture: For any $n > 2$ there is a prime $p < n$ with $p^{2} ϕ (n - p) - 1$ prime.

Let $ϕ$ denote Euler's totient function. The sequence $a (n)$ counts the primes $p < n$ such that $pϕ (n - p) - 1$ is also prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«233566» (2 statements).

theorem conjecture_1 (n : ℕ) (hn : 3 < n) : 0 < a n

theorem conjecture_2 (n : ℕ) (hn : 2 < n) :
    ∃ p < n, p.Prime ∧ Nat.Prime (p ^ 2 * (n - p).totient - 1)

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

A233566

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.