Erdős Problem #931 — open problem in number theory

Erdős Problem #931

Let k_1 ≥ k_2 ≥ 3. Are there only finitely many n_2≥ n_1 + k_1 such that Π_1≤ i≤ k_1(n_1 + i) and Π_1≤ j≤ k_2 (n_2 + j) have the same prime factors?

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-931,
  title        = {Erdős Problem #931},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-931}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $k_{1} \geq k_{2} \geq 3$ . Are there only finitely many $n_{2} \geq n_{1} + k_{1}$ such that $1 \leq i \leq k_{1} \prod (n_{1} + i) and 1 \leq j \leq k_{2} \prod (n_{2} + j)$ have the same prime factors?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«931». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_931 : answer(sorry) ↔ ∀ᵉ (k₁ : ℕ) (k₂ ≥ 3), k₂ ≤ k₁ →
    { (n₁, n₂) | n₁ + k₁ ≤ n₂ ∧
      (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors =
      (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors }.Finite

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/931. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_931.variants.additional_condition — Erdős thought perhaps if the two products have the same factors then n_2 > 2(n_1 + k_1).
erdos_931.variants.exists_prime — Erdős was unable to prove that if the two products have the same factors then there must exist a prime between n_1 and n_2.

References

erdosproblems.com/931

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.