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Level A · Machine-checkable Hard Number theory P-erdos-860

Erdős Problem #860

Let h(n) be such that, for any m≥ 1, in the interval (m,m+h(n)) there exist distinct integers a_i for 1≤ i≤ π(n) such that p_i| a_i, where p_i denotes the ith prime. Estimate h(n).

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

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-860,
  title        = {Erdős Problem #860},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-860}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Nobody has worked on this problem here yet

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  3. Paste its answer here — check it and publish it under your name

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.

or , ,

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let be such that, for any , in the interval there exist distinct integers for such that , where denotes the th prime.

Estimate .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«860».

theorem erdos_860 : (fun n ↦ (h n : ℝ)) =Θ[atTop] erdos_860.answer

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/860. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/860
  • [ErPo80] Erdős, P. and Pomerance, C., *Matching the natural numbers up to with distinct multiples in another interval*. Nederl. Akad. Wetensch. Indag. Math. 42 (1980), 147-161. PDF
  • [Er92c] Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. 15 (1992), 34-50. PDF

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.