Partitions n = x + y with x^2 + xy + y^2 prime

Conjecture: a(n) > 0 for all n > 1 with the only exception n = 8.

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-218585,
  title        = {Partitions n = x + y with x^2 + xy + y^2 prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-218585}},
  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

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: $a (n) > 0$ for all $n > 1$ with the only exception $n = 8$ .

conjecture_2. Conjecture: The number of primes of the form $n x + (n - x)^{2}$ with $0 < x < n /3$ is positive for $n > 12$ .

_Zak Seidov_, Sep 25 2013

For $n \geq 1$ , $a (n)$ is the number of ways to write $n$ as $x + y$ with $0 < x \leq y$ such that $x^{2} + x y + y^{2}$ is prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«218585» (2 statements).

theorem conjecture_1 (n : ℕ) (hn : 1 < n) (h8 : n ≠ 8) : 0 < a n

theorem conjecture_2 (n : ℕ) (hn : 12 < n) :
    ∃ x : ℕ, 0 < x ∧ 3 * x < n ∧ Nat.Prime (n * x + (n - x) ^ 2)

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

A218585
E. Hecke, "Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Zweite Mitteilung", Math. Z. 6 (1920), 11–51.

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.