Erdős Problem #458 — open problem in number theory

Erdős Problem #458

Let lcm(1, …, n) denote the least common multiple of 1, …, n. Let p_k be the k-th prime. Is it true that for all k ≥ 1, lcm(1, …, p_k+1-1) < p_k · lcm(1, …, p_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-458,
  title        = {Erdős Problem #458},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-458}},
  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 $lcm (1, \dots, n)$ denote the least common multiple of ${1, \dots, n}$ . Let $p_{k}$ be the $k$ -th prime. Is it true that for all $k \geq 1$ , $lcm (1, \dots, p_{k + 1} - 1) < p_{k} \cdot lcm (1, \dots, p_{k})$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«458». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_458 :
    answer(sorry) ↔ ∀ k : ℕ, lcm_upto ((k + 1).nth Prime - 1)
     < k.nth Prime * lcm_upto (k.nth Prime)

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/458. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/458

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.