Number of odd primes between $n^2$ and $(n+1)^2$ with $(n/p) = 1$

formal-conjectures · oeis · ams-11

$a(n)$ is the number of odd primes $p$ between $n^2$ and $(n+1)^2$ such that the Legendre symbol
$\left(\frac{n}{p}\right) = 1$.

Conjecture: $a(n) > 0$ for all $n > 0$.
- _Zhi-Wei Sun_, Dec 29 2012

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/A185150
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1211.1588
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/185150.lean
Local target: Corpus.OEIS185150.conjecture
Source SHA-256: 555fc0657312ff558d6d8c96792bc618cacbe36c7fcd5ba5666453c5ddb377b0
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