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

formal-conjectures · oeis · ams-11

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

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

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A069004
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/69004.lean
Local target: Corpus.OEIS69004.conjecture1
Source SHA-256: 9e1d067b3ce0ce256bbddf71eb643d5717454b8d288b03cbbfcefa25ad812068
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record