Number of times n^2 + s^2 is prime for positive integers s < n

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-69004,
  title        = {Number of times n^2 + s^2 is prime for positive integers s < n},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-69004}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Conjecture: $a (n) > 0$ for all $n > 1$ .

The sequence $a (n)$ counts the number of integers $s \in {1, \dots, n - 1}$ such that $n^{2} + s^{2}$ is prime: $a (n) = s = 1 \sum n - 1 [Prime (n^{2} + s^{2})]$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«69004».

theorem conjecture1 (n : ℕ) (hn : 1 < n) : 0 < a n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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.

Variants

conjecture2.variants.without_upper_bound — The part of the stronger conjecture that survives the counterexample to its upper bound: 5 a(n) ≥ π(n) for n > 1, together with the three claimed equalities…

References

A069004

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.