Partitions 2n + 1 = x + y with x^4 + y^4 prime

Conjecture: 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-218656,
  title        = {Partitions 2n + 1 = x + y with x^4 + y^4 prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-218656}},
  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: $a (n) > 0$ for all $n \geq 1$ .

conjecture_2. Conjecture: If $x^{4} + y^{4}$ in the definition of $a (n)$ is replaced by $x^{2} + y^{2}$ , then the count is positive for all $n \geq 1$ .

_Thomas Ordowski_, Nov 03 2012

conjecture_3. Conjecture: If $x^{4} + y^{4}$ in the definition of $a (n)$ is replaced by $x^{8} + y^{8}$ , then the count is positive for all $n \geq 1$ except for $2 n + 1 \in {7, 9, 55, 73, 75, 105}$ (i.e., $n \in {3, 4, 27, 36, 37, 52}$ ).

_Thomas Ordowski_, Nov 03 2012; exceptions noted by _Mauro Fiorentini_, Sep 22 2023

conjecture_4. Conjecture: If $x^{4} + y^{4}$ in the definition of $a (n)$ is replaced by $x^{16} + y^{16}$ , then the count is positive for all $n \geq 1$ except for $2 n + 1 \in {5, 9}$ (i.e., $n \in {2, 4}$ ).

_Thomas Ordowski_, Nov 03 2012; exceptions noted by _Mauro Fiorentini_, Sep 22 2023

For $n \geq 1$ , $a (n)$ is the number of ways to write $2 n + 1$ as $x + y$ with $0 < x < y$ such that $x^{4} + y^{4}$ is prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«218656» (4 statements).

theorem conjecture_1 (n : ℕ) (hn : 1 ≤ n) : 0 < a n

theorem conjecture_2 (n : ℕ) (hn : 1 ≤ n) :
    ∃ x : ℕ, 1 ≤ x ∧ x ≤ n ∧ Nat.Prime (x ^ 2 + (2 * n + 1 - x) ^ 2)

theorem conjecture_3 (n : ℕ) (hn : 1 ≤ n) (h_exc : n ∉ ({3, 4, 27, 36, 37, 52} : Set ℕ)) :
    ∃ x : ℕ, 1 ≤ x ∧ x ≤ n ∧ Nat.Prime (x ^ 8 + (2 * n + 1 - x) ^ 8)

theorem conjecture_4 (n : ℕ) (hn : 1 ≤ n) (h_exc : n ∉ ({2, 4} : Set ℕ)) :
    ∃ x : ℕ, 1 ≤ x ∧ x ≤ n ∧ Nat.Prime (x ^ 16 + (2 * n + 1 - x) ^ 16)

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

A218656

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.