Google DeepMind's AlphaProof Nexus Solved Nine Open Erdős Problems, Two of Them Open for 56 Years — and the Proofs Passed Peer Review in Science

Google DeepMind has published the results of AlphaProof Nexus, a Gemini-driven theorem-proving agent, in the journal Science after a full round of peer review. The system solved 9 of 353 open problems posed by mathematician Paul Erdős, two of which had stood for 56 years. Alongside the nine, the team proved 44 of 492 conjectures from the OEIS integer-sequence encyclopedia, settled one open problem in algebraic geometry, and sharpened a long-standing bound in min-max optimization.

The publication lands in the middle of a fight over how AI-generated mathematics should reach the public. OpenAI released more than 700 manuscript-style proofs on GitHub earlier this month, drawing a public boycott call from Terence Tao and complaints from mathematicians who said they had been scooped or handed an impossible reviewing burden; three of those manuscripts were withdrawn after obvious errors surfaced. DeepMind, by contrast, submitted its work to Science and waited months for external referees. Two mathematicians — from Carnegie Mellon University and the University of South Carolina — wrote a companion commentary in the same issue titled "AI for mathematics research has arrived."

AlphaProof Nexus runs on Gemini 3.1 Pro and works in a tight loop: the model drafts a proof in the formal language Lean, the Lean compiler checks it line by line, and any error messages are fed back for revision. Because a compiled Lean proof is machine-verified, there is no room for the system to assert something it has not proved — every result ships with a certificate that anyone can re-run from the team's public GitHub repository. On the hardest problems the agent can call a specialized prover, but the paper reports that all nine Erdős results were reproducible without it.

The two longest-standing conquests show the range. Erdős problem #12, posed in 1970, asks for an infinite set of integers in which no three chosen elements satisfy a particular divisibility relation while the set remains dense enough; the agent built it in blocks, using the Chinese remainder theorem together with sets that avoid three-term arithmetic progressions. Erdős problem #125, open since 1996, concerns sums of integers whose base-3 and base-4 digits are only 0s and 1s.

Cost is a striking part of the paper. DeepMind estimates a few hundred dollars of compute per solved problem — a scale that makes exhaustive attacks on curated problem lists economically feasible for the first time. The paper lists 20 authors, including Aja Huang, previously a core researcher on AlphaGo.

Peer review does not end the argument. The value of the 44 OEIS proofs and the two Erdős solutions will be judged by working number theorists over the coming months, and formal verification guarantees the correctness of the Lean statements, not that they say what mathematicians care about. But the episode sets a template the field has lacked: publish the machine-checkable artifacts, let compilers and referees do their jobs, and let the community audit both.

For research labs, the message is that mathematics is becoming a verifiable, benchmark-driven workload — and that credibility now depends on process as much as on results. For mathematicians, tools like AlphaProof Nexus look less like replacements and more like an exhaustively patient junior collaborator that never tires of a failed compile.