Erdős Problem #873 — open problem in number theory

Erdős Problem #873

Let A = a_1 < a_2 < … ⊆ ℕ and let F(A,X,k) count the number of i such that [a_i,a_i+1, … ,a_i+k−1] < X, where the left-hand side is the least common multiple. Is it true that, for every ε > 0, there exists some k such that F(A,X,k) < X^ε?

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-873,
  title        = {Erdős Problem #873},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-873}},
  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 $A = {a_{1} < a_{2} < \dots} \subseteq N$ and let $F (A, X, k)$ count the number of $i$ such that $[a_{i}, a_{i + 1}, \dots, a_{i + k - 1}] < X$ , where the left-hand side is the least common multiple. Is it true that, for every $ϵ > 0$ , there exists some $k$ such that $F (A, X, k) < X^{ϵ}$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«873». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_873 : answer(sorry) ↔ ∀ᵉ (a : ℕ → ℕ) (ε > (0 : ℝ)), 0 < a 0 → StrictMono a →
    ∃ k, ∀ X > 0, F a X k < (X^ε).toEReal

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/873. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/873

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.