Least k > 0 such that (k+1)(k+2)⋯(k+n) + 1 is prime

(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2, which is never prime. Hence a(4) = 0. Conjecture: a(n) = 0 if and only if n = 4.

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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-oeis-78729,
  title        = {Least k > 0 such that (k+1)(k+2)⋯(k+n) + 1 is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-78729}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

$(k + 1) (k + 2) (k + 3) (k + 4) + 1 = (k^{2} + 5 k + 5)^{2}$ , which is never prime. Hence $a (4) = 0$ . Conjecture: $a (n) = 0$ if and only if $n = 4$ .

The sequence $a (n)$ is the least positive integer $k$ such that $(k + 1) (k + 2) \dots (k + n) + 1$ is prime, if such $k$ exists; otherwise $a (n) = 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«78729».

theorem conjecture (n : ℕ) (hn : 0 < n) : a n = 0 ↔ n = 4

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

A078729

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.