Erdős (Erdos) Problem #711: status in 2026

Erdős Problem #711

Let f(n,m) be minimal such that (m,m+f(n,m)) contains distinct integers a_1,…,a_n with k | a_k for 1 ≤ k ≤ n. Prove that max_m f(n,m) ≤ n^1+o(1).

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of ₹1000.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #711 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_711.parts.i as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #711 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_711.parts.i`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 1 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-711,
  title        = {Erdős Problem #711},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-711}},
  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

Nobody has worked on this problem here yet

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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 $f (n, m)$ be minimal such that $(m, m + f (n, m))$ contains distinct integers $a_{1}, \dots, a_{n}$ with $k ∣ a_{k}$ for $1 \leq k \leq n$ . Prove that $max_{m} f (n, m) \leq n^{1 + o (1)}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«711».

theorem erdos_711.parts.i : ∃ error : ℕ → ℝ,
    error =o[atTop] (fun _ ↦ (1 : ℝ)) ∧ ∀ᶠ n in atTop, ∀ m, ∃ L,
      HasDivisibleDistinctTuple n m L ∧ (L : ℝ) ≤ Real.rpow n (1 + error n)

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/711. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/711
[ErPo80] Erdős, P. and Pomerance, C., Matching the natural numbers up to $n$ with distinct multiples in another interval (1980).
[Er92c] Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34–50.
[vD26] van Doorn, W., [*On the length of an interval that contains distinct multiples of the first $n$ positive integers*](https://arxiv.org/abs/2601.16972). Integers 26 (2026), #A7.

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.