DeepMind's AlphaProof Nexus agent autonomously resolved 9 of 353 formalized open Erdős problems at a few hundred dollars each.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
The result comes from a Google DeepMind paper posted to arXiv on 21 May 2026 (Tsoukalas, Kovsharov, Shirobokov and eighteen co-authors, "Advancing Mathematics Research with AI-Driven Formal Proof Search"). The team ran its most capable AlphaProof Nexus agent, a Gemini 3.1 Pro proving loop with Lean compiler feedback, evolutionary search and AlphaProof as a tool, on every Erdős problem then formalized in the open Formal Conjectures repository, 353 Lean statements out of a catalog of roughly 650 open problems, stopping each search after 3000 episodes. It reports complete Lean proofs for nine statements, at an inference cost it puts at a few hundred dollars per solved problem, and has published the proofs. The community wiki maintained under Terence Tao's Erdős problems project independently records the DeepMind prover agent's results and lists full solutions to problems 125, 152, 741 and 846 and new results on both parts of problem 12, all with non-significant human involvement. The core of the claim therefore stands on a primary source with public, machine-checked artifacts and an independent community record.
Three qualifications keep it short of established fact as worded. First, the nine are nine formal statements, not nine problems Erdős posed: the paper's own table marks the results on 138 and 26 as variants (the wiki records 138 as a partial result, and 26 as a stronger version of a problem Ruzsa had already solved), and the pairs 12(i)/(ii) and 741(i)/(ii) are parts of single catalog entries, so about six catalog problems are involved, of which four are fully resolved. Second, "autonomously" is fair by the wiki's standard, but for problems 125 and 741(i) humans corrected the formal statements after the agent first proved an easier reading based on natural density and then re-ran the agent, and experts checked each proved statement against the original conjecture afterwards. Third, the cost figure is self-reported and, as the paper says, varies widely and excludes the compute spent searching across all 353 problems.
The one dissenting source, a blog post arguing that at least four of the nine were already solved in the literature, misreads the wiki it cites: only problem 26 carries a full prior solution, and to a weaker statement than the one proved. What would move the claim to verified is independent confirmation of the cost figures or of the run logs; what would move it against is a prior solution to one of the four fully resolved problems, or evidence that proof ideas were supplied by humans.
Full reasoning: the evidence and decisions behind this verdict
Primary source: arXiv:2605.22763 (v1 21 May 2026, v2 8 June 2026), read in full for Sections 1, 2, 3, 5 and 6. Abstract and Section 3 state the result: agent (D) run on all 353 formal statements in the Formal Conjectures repository, 9 solved, experts validating afterwards that each Lean statement captured the original conjecture, proofs at www.github.com/google-deepmind/alphaproof-nexus-results. Table 1 lists 12(i), 12(ii), 125, 138*, 152, 741(i), 741(ii), 846, 26*†, with the asterisk marking variants and the dagger noting that 26 as proved was not posed by Erdős. Section 3 also records that the statements for 125 and 741(i) were amended from "density" to lower and upper density after the agent found proofs under natural density, and that the agent then proved the corrected statements. Section 5 ("Cost and Variance") states that per-problem costs have high variance and exclude the cost of identifying tractable problems across all 353; AlphaProof alone cost about 27.5 TPU hours (about 60 USD) per problem.
Independent corroboration: the wiki "AI contributions to Erdős problems" (github.com/teorth/erdosproblems, last updated 30 June 2026), read directly. Entries for the DeepMind prover agent: 125, full solution (Lean), 30 March 2026, section 1(a) standalone; 741, full solution (Lean), 16 April 2026, 1(a); 138, partial result (Lean), 10 April 2026, 1(a); 152, full solution (Lean), 3 April 2026, section 1(b) with partial literature Erdős, Sárközy and Sós (1994) marked similar; 846, full solution (Lean), 21 to 25 February 2026, 1(b), with partial literature Reiher, Rödl and Sales (2024) marked not similar, and an independent OpenAI solution; 12, section 1(c) building on literature (Erdős and Sárközy 1970, partial), outcome partial result (Lean) with solutions to the first and second parts; 26, section 1(c), literature Ruzsa (full), outcome solution to stronger problem. All of sections 1(a) to 1(c) carry the criterion "human involvement: non-significant".
Weighing the subclaims. The nine statements were unresolved in the literature when proved holds for 125, 741(i), 741(ii), 152, 846, 12(i), 12(ii) and 138* on the wiki's record; for 26* the base problem was solved by Ruzsa but the stronger statement proved was not, so it holds on a strict reading and fails on a loose one. The only human contribution was formalizing the statements is what the paper describes and what the wiki's section placement implies; the reformalization of 125 and 741(i) is work on statements, not proofs. Two of the nine are variants is the paper's own disclosure and qualifies the wording "9 open Erdős problems" without falsifying the count of formal statements. Problem 125's sumset has lower density zero is the flagship result, with a public Lean proof; the Lean artifacts were not re-checked here.
Instances: the paper affirms (primary); Quanta (3 August 2026) repeats the abstract verbatim, faithfully, and adds that the 353 were the formalized problems; the AIchats Substack post (31 May 2026) denies, but its central factual assertion (four of nine already solved) is contradicted by the wiki it relies on. The distribution of credible assertion is lopsided in favour, with the dissent reducing to the qualifications above rather than to a contrary fact.
Verdict: supported rather than verified because the cost figure rests on DeepMind's self-report, the run itself was not externally observed, and the phrase "9 open Erdős problems" overstates the number of distinct Erdős-posed problems resolved. Credence 0.85 that the claim is true as stated, reading "9 of 353 formalized open Erdős problems" as nine of the 353 formal statements. It would fall if a prior solution surfaced for 125, 152, 741 or 846, or if the cost were shown to be far higher; it would rise with independent access to run logs or cost accounting.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
The nine successes are nine Lean statements rather than nine catalog problems: because two of the nine are variants of catalog problems rather than problems Erdős posed, and two further pairs (12(i) and 12(ii); 741(i) and 741(ii)) are parts of single entries, the statements correspond to about six catalog problems; and to the extent that the nine statements were unresolved in the literature when proved fails for any of them, the count of open problems resolved falls further below nine.
The inference goes through as a qualification of the wording rather than a refutation: granting that two of the nine statements are variants, which the paper's own table discloses, the nine formal successes correspond to about six catalog problems, four of them fully resolved. It does not show the count of formal statements wrong, and it would only cut deeper if the nine statements were unresolved when proved failed for one of the fully resolved problems, which the community record does not currently indicate; the one genuine prior solution, Ruzsa's on problem 26, was to a weaker statement than the one proved.
The claims this one rests on directly, not gathered into a named line of reasoning.
- requiresa load-bearing premise: the parent is false without itsteward instructions →The only human mathematical contribution to AlphaProof Nexus's Erdős problem proofs was formalizing the problem statements. ↗︎
- supportsthis provides evidence for the parentsteward instructions →The sumset of the integers with only digits 0 and 1 in base 3 and those with only digits 0 and 1 in base 4 has lower density zero. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
Every affirmation of this result traces to one document, the Google DeepMind paper introducing AlphaProof Nexus, which the Quanta feature and the technology press quote without independent checking; the paper itself publishes the Lean proofs and, in its table of results, discloses that two of the nine statements are variants and that two pairs are parts of single catalog problems. The independent check is the community wiki on AI contributions to Erdős problems, which confirms full solutions to problems 125, 152, 741 and 846 and new results on the two parts of problem 12, while recording the problem 138 result as partial and the problem 26 result as a stronger version of a problem Ruzsa had solved. The one dissenting source, a blog post produced with a language model, misreads that wiki when it says four of the nine were already solved.
our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars
A May 2026 Google DeepMind paper; team narrowed search to problems written in formal logic.
The source's own evidence bears what it asserts. The magazine quotes the paper's abstract directly and adds its own useful caveat that the 353 problems were those already formalized in Lean, a fraction of the catalog's roughly 650 open entries. It offers no independent verification of the nine results and does not mention that two of the nine are variants.
Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research.
Abstract of the Google DeepMind paper introducing AlphaProof Nexus; the primary source for the result, reporting the agent's run on all 353 Lean-formalized Erdős problems in the Formal Conjectures repository.
The assertion outruns the source's own evidence. The paper backs the result with public Lean proofs and a table of the nine statements, and its own table discloses that two of the nine are variants of catalog problems, one not posed by Erdős, and that two pairs are parts of single problems. The abstract's phrase "9 open Erdős problems" thus reads more strongly than the body supports; the body's "9 Erdős problems out of 353 attempted" formal statements is the accurate count. The cost figure is stated by the paper itself to exclude the compute spent searching across all 353 problems and to vary widely between problems. Worth reading closely: Section 3 and Table 1 give the exact list of statements, the variant markings and the reformalization of problems 125 and 741(i); Section 5 gives the cost caveats. These are the details the headline omits.
DeepMind claimed they solved 9 “open” problems. The mathematical community audited them and found that at least four of them were already solved in human literature decades ago, and one is only a partial result.
A critical review of the DeepMind paper conducted as a conversation with Gemini 3.1, cross-referencing Table 1 with Tao's community wiki and arguing the "open problem" and "autonomous" descriptions are overstated.
The source's own material cuts against its assertion. The post's central assertion that at least four of the nine were already solved in the literature misreads the wiki it cites: the wiki marks the prior literature for problems 12, 152 and 846 as partial results, records the DeepMind proofs of 125, 152, 741 and 846 as full solutions, and records only problem 26 as having a full prior solution (by Ruzsa), to a weaker statement than the one proved. Its narrower points stand: the problem 138 result is a partial result for that problem, two of the nine are variants, and the statements for 125 and 741(i) were corrected by humans after the agent first proved an easier reading. The analysis was produced with a language model and is not independently sourced.
How these sources relate
- https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/ restates https://arxiv.org/abs/2605.22763, faithfully. Quanta quotes the abstract of arXiv:2605.22763 verbatim and attributes it to the DeepMind team of 21 researchers (the paper has 21 authors). The quotation is exact and the added parenthetical about formalized problems matches the paper's Section 3.
- https://aichats.substack.com/p/deepminds-alphaproof-nexus cites in support https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems, in a way that document does not support. The post reads the wiki's section headings as meaning the problems were already solved. The wiki itself marks the prior literature for 12, 152 and 846 with the partial-progress indicator and records the DeepMind results as full solutions (or, for 12, new solutions to the two parts); only 26 carries a full prior solution, and to a weaker statement. The dependency is real but the wiki does not support what the post takes from it.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.