Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 3

claims/: The 3 claim pages of Problem 3, one per claimant's result; the problem's standing derives from them.

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

Status. 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 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, and the case $A$ the set of primes, proved by Green and Tao, is an accepted partial result on its own claim page.

Source. erdosproblems.com/3, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #3, https://www.erdosproblems.com/3.

References.

[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_unsolved_problems_number_theory.
[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).

Formalization. Statement in formal-conjectures at its commit of 2026-10-06, which states the question as erdos_3 with answer(sorry) and a sorry body and carries no formal_proof attribute; its five solved variants are the three-term case and the $r_k$ bounds of Kelley--Meka, Green--Tao, Gowers and Leng--Sah--Sawhney recorded in the Current assessment.

Current assessment

The site's formulation (page last edited 4 April 2026) asks whether every $A\subseteq\mathbb N$ with divergent reciprocal sum contains arbitrarily long arithmetic progressions. The answer is yes by Corollary 1.2 of the OpenAI release manuscript of 23 September 2026, accepted on its claim page on the release's Lean declaration, which this corpus built and whose axioms it checked; the site's label is OPEN and the release has no journal record or outside review.

Known results before the release, as the site's commentary and the cited papers give them. The case $k=3$ is Corollary 1.2 of Bloom and Sisask [BlSi20], deduced by partial summation from their bound $r_3(N)\ll N/(\log N)^{1+c}$ ; it is the claimed partial result on its claim page, with no journal version. Kelley and Meka [KeMe23] proved $r_3(N)\ll N\exp(-c(\log N)^{1/12})$ , which gives the three-term case with room to spare. Green and Tao [GrTa17] proved $r_4(N)\ll N/(\log N)^c$ for some $c>0$ . Gowers [Go01] proved $r_k(N)\ll N/(\log\log N)^{c_k}$ for every $k$ , and Leng, Sah and Sawhney [LSS24] improved this for every $k\ge5$ to $r_k(N)\ll N/\exp((\log\log N)^{c_k})$ ; neither bound is summable over dyadic blocks, so neither settles a case $k\ge5$ , and neither has a claim page. Green and Tao's theorem that the primes contain arbitrarily long arithmetic progressions [GrTa08] is the special case $A$ the set of primes, proved directly; Erdős had viewed this problem as the only way to approach it. It is an accepted partial result on its own claim page. The formal-conjectures file records the three-term case and the four $r_k$ bounds as its solved variants.

Search scope: the site's page as accessed 2026-09-04 and 2026-10-06, its references, the formal-conjectures statement file at its commit of 2026-10-06, and the OpenAI release of 23 September 2026 at the revision pinned on the claim page.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (45)

Green and Tao's progressions of primes Bloom and Sisask's three-term case OpenAI's proof of the reciprocal-sum conjecture problems/additive_combinatorics/E0003/claims OpenAI's quasipolynomial bound for progression-free sets OpenAI's stretched-exponential bound at k = 3, with Lean Problem 169 Gerver's lower bound f(k) at least (1-o(1)) k log k OpenAI's uniform bound on reciprocal sums of progression-free sets Problem 201 Problem 219 OpenAI's reciprocal-sum theorem applied to the primes Sumsets and Arithmetic Progressions Sumsets and Arithmetic Progressions Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions Corollary 1.2 (p. 2): a set with divergent reciprocal sum contains infinitely many three-term progressions Theorem 1.1 (p. 1): a three-term-progression-free subset of {1,...,N} has size << N/(log N)^{1+c}additive_combinatorics/brown_1990_quasi_progressions_descending_waves Theorem 2 (p. 5): Erdős's progression conjecture is equivalent to its quasi-progression version Theorem 3 (p. 5): a set with infinite reciprocal sum contains arbitrarily large cubes additive_combinatorics/erdos_1979_old_new_problems_results_combinatorial_number additive_combinatorics/green_2008_primes_contain_arbitrarily_long_arithmetic_progressions Conjecture 2.2 (p. 3): Erdős's conjecture that a sequence with divergent reciprocal sum contains arbitrarily long progressions Theorem 1.1 (p. 2): the primes contain infinitely many k-term arithmetic progressions for every k additive_combinatorics/green_2017_new_bounds_szemeredi_s_theorem Theorem 1.1 (p. 2): r_4(N) << N(log N)^{-c}additive_combinatorics/grosswald_1982_arithmetic_progressions_that_consist_only_primes Corollary (p. 12): if the conditional formula (2) holds, the primes contain arbitrarily long arithmetic progressions additive_combinatorics/kelley_2023_strong_bounds_3_progressions additive_combinatorics/leng_2024_improved_bounds_szemeredi_s_theorem Theorem 1.1 (p. 1): r_k(N) << N exp(-(log log N)^{c_k}) for every fixed k >= 5 additive_combinatorics/openai_2026_quasipolynomial_bounds_arithmetic_progressions Corollary 11.2: a uniform bound on reciprocal sums of k-progression-free sets Corollary 1.2: divergent reciprocal sum forces progressions of every length Theorem 1.1: r_k(N) ≤ C_k N exp(-c_k (log N)^{ε_k}) for every fixed k ≥ 3 discrete_geometry/graham_2017_euclidean_ramsey_theory Conjecture 11.5.6 (p. 291): Erdős's conjecture on sets with divergent reciprocal sum distance_problems/erdos_1983_combinatorial_problems_geometry Conjecture (p. 44): a sequence of integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions distance_problems/graham_2004_euclidean_ramsey_theory Conjecture 11.5.6 (p. 10): Erdős's conjecture that divergence of the sum of reciprocals forces arbitrarily long arithmetic progressions number_theory/guy_2004_unsolved_problems_number_theory ramsey_theory/erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk Conjecture (p. 156, unnumbered): a sequence with divergent sum of reciprocals contains arbitrarily long arithmetic progressions set_systems/erdos_1981_combinatorial_problems_which_i_would_most

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