Erdős (Erdos) Problem #388: status in 2026

Erdős Problem #388

Are there only finitely many solutions of Π_1≤ i≤ k_1(m_1+i)=Π_1≤ j≤ k_2(m_2+j) with k_1,k_2>3 and m_1+k_1≤ m_2? Here m_1, m_2 ≥ 0, so every factor is positive and the condition m_1 + k_1 ≤ m_2 makes the two blocks disjoint.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #388 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_388 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #388 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_388`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). 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-erdos-388,
  title        = {Erdős Problem #388},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-388}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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

Are there only finitely many solutions of $1 \leq i \leq k_{1} \prod (m_{1} + i) = 1 \leq j \leq k_{2} \prod (m_{2} + j)$ with $k_{1}, k_{2} > 3$ and $m_{1} + k_{1} \leq m_{2}$ ?

Here $m_{1}, m_{2} \geq 0$ , so every factor is positive and the condition $m_{1} + k_{1} \leq m_{2}$ makes the two blocks disjoint. Negative $m_{1}$ would give trivial solutions such as $(- 5) (- 4) (- 3) (- 2) = 2 \cdot 3 \cdot 4 \cdot 5$ . The source also asks to classify all solutions; see Erdos388.erdos_388.variants.classification.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«388».

theorem erdos_388 :
    {(m₁, k₁, m₂, k₂) : ℕ × ℕ × ℕ × ℕ | 3 < k₁ ∧ 3 < k₂ ∧ m₁ + k₁ ≤ m₂ ∧
      ∏ i ∈ Finset.Icc 1 k₁, (m₁ + i) = ∏ j ∈ Finset.Icc 1 k₂, (m₂ + j)}.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/388. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_388.variants.classification — Can one classify all solutions of Π_1≤ i≤ k_1(m_1+i)=Π_1≤ j≤ k_2(m_2+j) with k_1,k_2>3 and m_1+k_1≤ m_2?
erdos_388.variants.general — More generally, if k_1>2 then for fixed a and b the equation aΠ_1≤ i≤ k_1(m_1+i)=bΠ_1≤ j≤ k_2(m_2+j) should have only finitely many solutions.

References

erdosproblems.com/388
[Er76d] Erdős, P., *Problems and results on number theoretic properties of consecutive integers and related questions*. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.
[ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number theory*. Monographies de L'Enseignement Mathematique (1980).
[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.

Source and licence

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