Erdős Problem #3 — open problem in number theory

Erdős Problem #3

If A ⊂ ℕ has Σ_n ∈ Afrac 1 n = ∞, then must A contain arbitrarily long arithmetic progressions?

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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-erdos-3,
  title        = {Erdős Problem #3},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-3}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

If $A \subset N$ has $\sum_{n \in A} \frac{1}{n} = \infty$ , then must $A$ contain arbitrarily long arithmetic progressions?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«3». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_3 : answer(sorry) ↔ ∀ A : Set ℕ,
    (¬ Summable fun a : A ↦ 1 / (a : ℝ)) →
    ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength k

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/3. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/3
[BlSi20] Bloom, Thomas F. and Sisask, Olof, *Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions*. arXiv:2007.03528 (2020).
[Go01] Gowers, W. T., A new proof of Szemerédi's theorem. Geom. Funct. Anal. (2001), 465-588.
[GrTa17] Green, Ben and Tao, Terence, *New bounds for Szemerédi's theorem, III: a polylogarithmic bound for $r_{4} (N)$ *. Mathematika (2017).
[KeMe23] Kelley, Zander and Meka, Raghu, Strong bounds for 3-progressions. 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) (2023).
[LSS24] Leng, James, Sah, Ashwin and Sawhney, Mehtaab, *Improved bounds for Szemerédi's theorem*. arXiv:2402.17995 (2024).

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.