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Level A · Machine-checkable Hard Number theory P-erdos-1210

Erdős Problem #1210

Let A⊆ [1,n) be a set of integers such that (a,b)=1 for all distinct a,b∈ A. Is it true that Σ_a∈ A1/n-a≤ Σ_p < n1/p+O(1)?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-04-04).

Current state (tracked automatically)

  • erdosproblems.com lists Erdős problem #1210 as open (status updated 4 April 2026).
  • Formal Conjectures marks the Lean statement erdos_1210 as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #1210 on erdosproblems.comopen
2026-04-04
erdosproblems.com
Formal Conjectures: status of `erdos_1210`research openFormal Conjectures
Formal Conjectures: research statements in the file2 open, 0 solvedFormal 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-1210,
  title        = {Erdős Problem #1210},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1210}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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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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Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  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 , ,

ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.

Current state

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

The problem

The question

Let be a set of integers such that for all distinct . Is it true that ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1210».

theorem erdos_1210 :
    ∃ C : ℝ, ∀ n : ℕ, ∀ A : Finset ℕ,
      (∀ a ∈ A, 1 ≤ a ∧ a < n) →
      (∀ a ∈ A, ∀ b ∈ A, a ≠ b → a.Coprime b) →
      ∑ a ∈ A, (1 / ((n : ℝ) - a)) ≤ (∑ p ∈ (range n).filter Prime, (1 / (p : ℝ))) + C

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/1210. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_1210.variants.er80_correction — In [Er80] he claims he "did not state this quite correctly" in [Er77c].

References

  • erdosproblems.com/1210
  • [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
  • [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

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.