Erdős Problem #677 — open problem in number theory

Erdős Problem #677

Denote by M(n, k) the least common multiple of the finite set n+1, dotsc, n+k. Is it true that for all m ≥ n + k, we get M(m, k) ≠ 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-677,
  title        = {Erdős Problem #677},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-677}},
  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

Denote by $M (n, k)$ the least common multiple of the finite set ${n + 1, \dots, n + k}$ . Is it true that for all $m \geq n + k$ , we get $M (m, k) \neq = M (n, k)$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«677».

theorem erdos_677 :
    ∀ (m n k : ℕ), k > 0 → m ≥ n + k → lcmInterval m k ≠ lcmInterval n k

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/677. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/677

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.