Erdős Problem #930 — open problem in number theory

Erdős Problem #930

Is it true that, for every r, there is a k such that if I_1,…,I_r are disjoint intervals of consecutive integers, all of length at least k, then Π_1≤ i≤ rΠ_m∈ I_im is not a perfect power?

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-930,
  title        = {Erdős Problem #930},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-930}},
  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

Is it true that, for every $r$ , there is a $k$ such that if $I_{1}, \dots, I_{r}$ are disjoint intervals of consecutive integers, all of length at least $k$ , then $1 \leq i \leq r \prod m \in I_{i} \prod m$ is not a perfect power?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«930». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_930 :
    answer(sorry) ↔ ∀ r > 0, ∃ k, ∀ I₁ I₂ : Fin r → ℕ,
      (∀ i : Fin r, 0 < I₁ i ∧ I₁ i + k ≤ I₂ i + 1) →
        (∀ i j : Fin r, i < j → I₂ i < I₁ j) →
          ¬ IsPower (∏ i : Fin r, ∏ m ∈ Icc (I₁ i) (I₂ i), m)

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/930. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/930

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.