Number of pairs of twin primes between $n^2$ and $(n+1)^2$

formal-conjectures · oeis · ams-11

$a(n)$ is the number of pairs of twin primes between $n^2$ and $(n+1)^2$.
This counts the number of primes $p$ such that $p$ and $p+2$ are both prime,
and the entire twin prime pair $(p, p+2)$ lies strictly between $n^2$ and $(n+1)^2$.
That is, $n^2 < p$ and $p + 2 < (n+1)^2$.

It is conjectured that $a(n)>0$ for all $n>122$.
Proving this would also prove Legendre's conjecture that there is a prime
between $n^2$ and $(n+1)^2$. - _T. D. Noe_, Feb 28 2007

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/A091591
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/91591.lean
Local target: Corpus.OEIS91591.conjecture
Source SHA-256: d431f49901cd37c4b5171ca0f5d093501a5a4792197b51f766e07accb87783ca
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.

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