Erdős Problem #686 — open problem in number theory

Erdős Problem #686

Can every integer N≥2 be written as N=Π_1≤ i≤ k(m+i)/Π_1≤ i≤ k(n+i) for some k≥2 and m≥n+k?

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-686,
  title        = {Erdős Problem #686},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-686}},
  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

Can every integer $N \geq 2$ be written as $N = \frac{\prod _{1 \leq i \leq k} ( m + i )}{\prod _{1 \leq i \leq k} ( n + i )}$ for some $k \geq 2$ and $m \geq n + k$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«686». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_686 :
    answer(sorry) ↔ ∀ N ≥ (2 : ℕ), ∃ᵉ (k ≥ 2) (n : ℕ) (m ≥ n + k),
      (N : ℚ) = (∏ i ∈ Finset.Icc 1 k, (m + i)) / (∏ i ∈ Finset.Icc 1 k, (n + i))

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/686. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_686.variants.square — Can every square N≥2 be written as N=Π_1≤ i≤ k(m+i)/Π_1≤ i≤ k(n+i) for some k≥2 and m≥n+k?
erdos_686.variants.four — Can 4 be written as 4=Π_1≤ i≤ k(m+i)/Π_1≤ i≤ k(n+i) for some k≥2 and m≥n+k?
erdos_686.variants.twenty_five — Can 25 be written as 25=Π_1≤ i≤ k(m+i)/Π_1≤ i≤ k(n+i) for some k≥2 and m≥n+k?

References

erdosproblems.com/686
[Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.

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.