a(n) = the number of values of k <= 10^n such that √(k(k+1)(k+2)(k+3)+1) is prime

Conjecture: a(n)/A006880(n) → 1.77... where A006880(n) is the number of primes ≤ 10^n.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement conjecture as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `conjecture`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-11). How it works.

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-115366,
  title        = {a(n) = the number of values of k <= 10^n such that √(k(k+1)(k+2)(k+3)+1) is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-115366}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Conjecture: $a (n) / A 006880 (n) \to 1.77...$ where A006880(n) is the number of primes $\leq 1 0^{n}$ .

Since $k (k + 1) (k + 2) (k + 3) + 1 = k^{2} + 3 k + 1$ , $a (n) = # {k \in N ∣ 1 \leq k \leq 1 0^{n} \land (k^{2} + 3 k + 1) is prime} .$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«115366».

theorem conjecture :
    ∃ L : ℝ,
      Tendsto (fun n : ℕ => (a n : ℝ) / (Nat.primeCounting' (10 ^ n) : ℝ)) atTop (nhds L) ∧
      1.77 ≤ L ∧ L ≤ 1.78

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

A115366

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.