DeepMind's AlphaProof Nexus agent autonomously resolved 9 of 353 formalized open Erdős problems at a few hundred dollars each.
4 events · 1 assessment · 2 decisions
Structured and assessed
First pass. Read the Quanta source whole, located and read the primary source (arXiv:2605.22763, Sections 1 to 6), read Tao's community wiki on AI contributions to Erdős problems as the independent record, and read the one dissenting source (AIchats Substack). Decomposed into four new subclaims (all Matcher-confirmed novel): openness of the nine statements (requires), human involvement limited to formalization (requires), two of nine being variants (contradicts, grouped with openness under an against-argument "Headline count qualifications"), and the problem 125 mathematical result (supports, mathematical type, deferred stub). Minted the parent "AI systems have autonomously resolved open Erdős problems" (supports), which the Matcher found absent though several specific-case claims exist; linked laterally to the OpenAI unit distance autonomy claim as the parallel announcement. Recorded the arXiv paper as an affirming instance and the Substack post as a denying instance (confidence 0.55, LLM-assisted critique). Provenance: readings for all three instances, Quanta repeats the paper faithfully, the paper's abstract overstates relative to its own table, the Substack post misreads the wiki it cites; map written and marked material because all affirmations trace to one document and the independent check carries the qualifications. Canonical form updated to name AlphaProof Nexus, since DeepMind published two distinct Erdős papers in 2026. Importance kept at 0.4 (notable, moderately contested). Assessed supported, confidence 0.8, credence 0.85: primary source plus public Lean proofs plus independent wiki confirm the core; self-reported cost, unobserved run, and the variants/parts issue keep it from verified. No dependents to notify beyond the newly minted parent, which will be assessed by its own Steward.
Assessed Supported
verdict confidence 0.80 · credence 0.85
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.
Add parent claim
Minted parent claim 56e32774-f9c0-4d78-9655-0c46ec9068e9 ("AI systems have autonomously resolved open Erdős problems.") and attached this claim as its subclaim (supports): This claim (DeepMind's AlphaProof Nexus resolving nine formalized Erdős statements) is one of several specific results the discourse cites as evidence that AI has autonomously resolved open Erdős problems; the graph holds parallel specific-case claims (OpenAI's unit distance disproof, a165bdad-95cf-48c6-8b28-ebce02d15cfa; early AI-driven Erdős solutions from hobbyists, b54f93ba-30e2-4b94-8375-5f52132fa258) but the Matcher found no node for the general proposition they all bear on. The general claim is what Quanta, Tao's wiki and the labs' announcements argue about.
Claim entered the graph