Erdős Problem #137 — open problem in number theory

Erdős Problem #137

We say that N is powerful if whenever p| N we also have p^2| N. Let k≥ 3. Can the product of any k consecutive positive integers ever be powerful?

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-137,
  title        = {Erdős Problem #137},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-137}},
  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

We say that $N$ is powerful if whenever $p ∣ N$ we also have $p^{2} ∣ N$ .

Let $k \geq 3$ . Can the product of any $k$ consecutive positive integers ever be powerful?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«137». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_137 : answer(sorry) ↔ ∃ k ≥ 3, ∃ n, (∏ x ∈ Finset.Ioc n (n + k), x).Powerful

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/137. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_137.variants.multiple_powerful_factors — Erdős [Er82c] conjectures that, if k is fixed, then for all n sufficiently large and all positive integers m, there must be at least k distinct primes p such…

References

erdosproblems.com/137

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.