Partitions of n into x + y with (xy)^2 + 1 prime

Conjecture: a(n) > 0 if n is different from 1, 6, 16, 24.

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-219791,
  title        = {Partitions of n into x + y with (xy)^2 + 1 prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-219791}},
  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: $a (n) > 0$ if $n$ is different from $1, 6, 16, 24$ .

conjecture_2. Conjecture: For any positive integer $k$ , each sufficiently large integer $n$ can be written as $x + y$ ( $x > 0, y > 0$ ) with $(x y)^{2^{k}} + 1$ prime.

_Zhi-Wei Sun_

Number of ways to write $n = x + y$ ( $0 < x \leq y$ ) with $(x y)^{2} + 1$ prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«219791» (2 statements).

theorem conjecture_1 (n : ℕ) (hn : n > 0) (h1 : n ≠ 1) (h6 : n ≠ 6) (h16 : n ≠ 16) (h24 : n ≠ 24) :
    a n > 0

theorem conjecture_2 (k : ℕ) (hk : 0 < k) :
    ∃ N : ℕ, ∀ n : ℕ, N ≤ n →
      ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ x + y = n ∧ Nat.Prime ((x * y) ^ (2 ^ k) + 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

A219791

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.