Erdős (Erdos) Problem #291: status in 2026

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?

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

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #291 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_291.parts.i as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #291 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_291.parts.i`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 open, 3 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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.

The problem

The question

Let $n \geq 1$ and define $L_{n}$ to be the least common multiple of ${1, \dots, n}$ and $a_{n}$ by $\sum_{1 \leq k \leq n} \frac{1}{k} = \frac{a _{n}}{L _{n}}$ .

Is it true that $(a_{n}, L_{n}) = 1$ occurs for infinitely many $n$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«291».

theorem erdos_291.parts.i :
    { 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.