Problem 3

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

Source: erdosproblems.com/3

Number theory
Additive combinatorics
Arithmetic progressions

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SolvedProved

An accepted solution exists. The statement is true.

OPEN, the site's label (page last edited 4 April 2026, as accessed 2026-09-04). The OpenAI mathematics release of 23 September 2026 answers the question yes. Its manuscript states $r_k(N)\le C_kN\exp(-c_k(\log N)^{\varepsilon_k})$ for every fixed $k\ge3$ (Theorem 1.1) and sums that bound over dyadic intervals (Corollary 1.2). The release's Lean proves the yes answer from a weaker bound, $r_k(N)\le CN\exp(-c(\log\log N)^{1+\eta})$ . This corpus built that declaration, checked its axioms and found it identical to the release's comparator challenge, so the result is accepted on its claim page (OpenAI, 2026) and the problem stands solved and proved here. Theorem 1.1's bound is not formalized; it is a claimed partial result on Problem 142's claim page. The case $k=3$ , proved by Bloom and Sisask, is a claimed partial result on its own claim page (Bloom and Sisask, 2020), and the case $A$ the set of primes, proved by Green and Tao, is an accepted partial result on its own claim page (Green and Tao, 2004).

BlSi20: Bloom, T.F. and Sisask, O., Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. arXiv:2007.03528 (2020).
Er81: Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
Er83c: Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.
Go01: Gowers, W. T., A new proof of Szemerédi's theorem. Geom. Funct. Anal. (2001), 465-588.
GrTa08: Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) (2008), 481-547.
GrTa17: Green, Ben and Tao, Terence, New bounds for Szemerédi's theorem, III: a polylogarithmic bound for $r_4(N)$ . Mathematika (2017), 944-1040.
Gu04: Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section A5 "Arithmetic progressions of primes", printed p. 25: "More generally, Erdős conjectures that if $\{a_i\}$ is any infinite sequence of integers for which $\sum1/a_i$ is divergent, then the sequence contains arbitrarily long arithmetic progressions. He offered $3000.00 for a proof or disproof of this conjecture"; the printed prize differs from the site's. Section E10, printed p. 318, points back to "a potentially remunerative conjecture of Erdős, which, if true, would imply Szemerédi's theorem". Library home: Guy 2004.
KeMe23: Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).
LSS24: Leng, J., Sah, A. and Sawhney, M., Improved bounds for Szemerédi's theorem. arXiv:2402.17995 (2024).

Problem 3

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