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Level A · Machine-checkable Hard Number theory P-erdos-291

Erdős Problem #291

Let n≥ 1 and define L_n to be the least common multiple of 1,…,n and a_n by Σ_1≤ k≤ n1/k=a_n/L_n. Is it true that (a_n,L_n)=1 occurs for infinitely many n?

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-291,
  title        = {Erdős Problem #291},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-291}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
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Verified
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Disputed
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Refuted
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On the literature board
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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 and define to be the least common multiple of and by .

Is it true that occurs for infinitely many ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«291». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_291.parts.i :
    answer(sorry) ↔
      { n : ℕ | Nat.gcd (a n) (L n) = 1 }.Infinite

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/291. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_291.variants.shiu_heuristic_asymptotic — This leads to a heuristic prediction (see for example a preprint of Shiu [Sh16]) of asympx/log x for the number of n∈ [1,x] such that (a_n,L_n)=1.
  • erdos_291.variants.shiu_heuristic_density_zero — In particular, there should be infinitely many n, but the set of such n should have density zero.

References

  • erdosproblems.com/291
  • [ErGr80, p.34] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
  • [Sh16] P. Shiu, The denominators of harmonic numbers. arXiv:1607.02863 (2016).
  • [WuYa22] Wu, Bing-Ling and Yan, Xiao-Hui, On the denominators of harmonic numbers. {IV}. C. R. Math. Acad. Sci. Paris (2022), 53--57.

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.