Erdős Problem #1210 — open problem in number theory

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)?

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 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $A \subseteq [1, n)$ be a set of integers such that $(a, b) = 1$ for all distinct $a, b \in A$ . Is it true that $\sum_{a \in A} \frac{1}{n - a} \leq \sum_{p < n} \frac{1}{p} + O (1)$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1210». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1210 :
  answer(sorry) ↔
    ∃ 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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.