Number of pairs of twin primes between n^2 and (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

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-91591,
  title        = {Number of pairs of twin primes between n^2 and (n+1)^2},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-91591}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

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The problem

The question

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

$a (n)$ 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 $p + 2 < (n + 1)^{2}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«91591».

theorem conjecture :
    ∀ n : ℕ, n > 122 → a n > 0

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.

References

A091591

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.