DeepMind's AlphaProof Nexus solves nine open Erdős problems, peer-reviewed in Science
DeepMind's AlphaProof Nexus solved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures; the work cleared peer review at Science on October 8.
AlphaProof Nexus, Google DeepMind’s AI system for research-level mathematics, has a peer-reviewed paper: “Advancing mathematics research with AI-driven formal proof search,” published in Science on October 8 (volume 394, issue 6820, pp. 234–239). Per the abstract, the system autonomously resolved 9 of 353 open Erdős problems — two of them unsolved for 56 years — and proved 44 of 492 conjectures from the OEIS integer-sequence encyclopedia. Gemini 3.1 Pro drafts each proof in the formal language Lean; a compiler checks every step and feeds the error messages back for the next draft.
Key points
- The result: 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures, published in Science vol. 394, issue 6820, on October 8, 2026.
- The bare-bones setup sufficed: a basic agent running a commercial LLM against the Lean compiler — no DeepMind-specific AlphaProof tooling — also solved all nine, it just spent more compute on the hardest ones.
- Everything is checkable: the Lean proofs and matching natural-language write-ups are open in the google-deepmind/alphaproof-nexus-results repo (Apache-2.0 and CC-BY 4.0); the preprint put inference at roughly a few hundred dollars per problem.
Background
The Erdős problems are the conjecture list mathematician Paul Erdős left behind, spanning combinatorics, number theory, graph theory and geometry — erdosproblems.com tracks them one by one. OEIS, the On-Line Encyclopedia of Integer Sequences, hoards its own pile of conjectures nobody has closed.
The results themselves are not new. The arXiv preprint (2605.22763) posted them on May 21, and IT Home walked through the system’s four-tier agent architecture on May 26. What October 8 adds is the academic stamp: the same body of work went through Science’s review, with the solutions posted online ahead of publication.
The timing supplies a pointed contrast. On October 6, OpenAI dumped 722 AI-generated math manuscripts straight onto GitHub, no journal involved — and an independent IAS advisory group had publicly asked OpenAI to stop exactly this kind of testing (see our earlier coverage: OpenAI publishes 722 AI math papers). One week, two very different routes to the same field.
The facts
The checkable numbers, from the paper and the preprint:
| Metric | Number | Source |
|---|---|---|
| Erdős problems | 353 attempted, 9 resolved | Paper abstract (verified via Crossref metadata) |
| Longest-standing pair | 56 years open | The Decoder, IT Home |
| OEIS conjectures | 492 attempted, 44 proved | Paper abstract |
| Other results | One 15-year-old Hilbert function problem; improved known bound in convex optimization | IT Home (from the preprint) |
| Inference cost | A few hundred dollars per problem | Preprint abstract |
| Publication | Science, vol. 394, issue 6820, pp. 234–239, 2026-10-08 | Crossref |
The working loop is simple to describe: Gemini 3.1 Pro writes a proof in Lean, the compiler verifies each step against the axioms, and any error message goes back to the model for the next attempt. The preprint ranks four agents by increasing capability. Agent A is just the model plus the compiler; Agent B adds DeepMind’s dedicated AlphaProof tool to patch gaps; Agent C layers in AlphaEvolve-style evolution, letting candidate proofs share, score and rank each other; Agent D has it all. The team expected the full stack to be needed for the Erdős set. Agent A cleared it anyway, paying more compute on the hardest problems. The paper’s abstract states it flatly: “Even a basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes.”
Per the abstract, the system is already deployed in research workflows across combinatorics, optimization, graph theory, algebraic geometry and quantum optics.
What others say
A companion commentary in the same issue of Science set the frame: the era of AI in research-level mathematics has arrived (per Science and Technology Daily).
The same report carries the worry. Mathematicians quoted by the outlet fear that AI at scale will erode the incentive to explore problems and actually understand proofs. The team’s answer: even failed AI attempts help mathematicians understand a problem better and spot new angles.
The Decoder puts this paper next to OpenAI’s release in one line: “DeepMind put its work through peer review.” The backlash against OpenAI’s unreviewed dump was covered by the New York Times and others (see also our October 7 report).
How to try it
The results live at github.com/google-deepmind/alphaproof-nexus-results. To verify the proofs yourself, the README gives two commands:
lake exe cache get
lake build
Note the repo’s own caveat: it contains only problems where successful proofs were discovered. The full attempt list is in erdos_problems_attempted.txt, and formalizations of the unsolved ones sit in the sibling formal-conjectures repo. The natural-language proofs are CC-BY 4.0, so you can cite and reuse them directly.
Our take
For anyone building agents, the most useful line in this paper is not the nine problems — it is the word “replicated” in the abstract. Put a deterministic verifier in the loop and a general-purpose model is enough; the evolution machinery and the specialized tooling were never shown to be necessary. Harder feedback signals make steadier agents, and the same lesson holds for coding agents.
The honest side: 9 out of 353 is under a 3% hit rate; the repo collects successes only, with the failures parked elsewhere; and the numbers have been public since May, so the increment here is academic recognition, not a new breakthrough. One thing the paper does not detail is how deeply reviewers actually checked the Lean proofs — that decides how much weight “peer-reviewed AI proofs” should carry, and it is worth watching.