Erdős Problem #12 — open problem in number theory

Erdős Problem #12

Let A be an infinite set such that there are no distinct a,b,c ∈ A such that a | (b+c) and b,c > a. Is it true that ∑_n ∈ A 1/n < ∞?

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-12,
  title        = {Erdős Problem #12},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-12}},
  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 $A$ be an infinite set such that there are no distinct $a, b, c \in A$ such that $a ∣ (b + c)$ and $b, c > a$ . Is it true that $\sum_{n \in A} \frac{1}{n} < \infty$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«12». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_12.parts.iii :
    answer(sorry) ↔ ∀ (A : Set ℕ), IsGood A → Summable (fun (n : A) ↦ (1 / n : ℝ))

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/12. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/12

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.