Erdős Problem #727 — open problem in number theory

Erdős Problem #727

Let k ≥ 2. Does ((n+k)!)^2∣(2n)! hold for infinitely many n?

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-erdos-727,
  title        = {Erdős Problem #727},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-727}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $k \geq 2$ . Does $((n + k)!)^{2} ∣ (2 n)!$ hold for infinitely many $n$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«727». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_727 : answer(sorry) ↔ ∀ k ≥ 2,
    Set.Infinite {n : ℕ | (Nat.factorial (n + k)) ^ 2 ∣ Nat.factorial (2 * n)}

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 — check erdosproblems.com/727. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_727.variants.k_2 — It is open even for k = 2. Let k = 2. Does ((n+k)!)^2∣(2n)! hold for infinitely many n?

References

erdosproblems.com/727

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.