The announcement
On October 8, 2026, Science published Google DeepMind's peer-reviewed paper on AlphaProof Nexus ("Advancing mathematics research with AI-driven formal proof search"). The preprint had been on arXiv since May 21. The system uses LLM-based agents that search for math proofs written in Lean, a language in which the computer mechanically checks every step.
The results
Per the paper:
- 9 of 353 attempted Erdős problems solved autonomously (~2.5%), including two open for 56 years (#12(i) and #12(ii), posed by Erdős and Sárközy in 1970). Inference cost: a few hundred dollars per problem.
- 44 of 492 conjectures from the On-Line Encyclopedia of Integer Sequences (OEIS).
- Resolved a 15-year-old question on Hilbert functions (algebraic geometry), improved a bound in convex optimization, and helped resolve a problem from Ben Green's list.
- Found misformalizations in the literature.
The Lean proofs are public on GitHub in google-deepmind/alphaproof-nexus-results (Apache 2.0, ~300 stars), and results were logged on Terence Tao's wiki of AI contributions to Erdős problems.
How it works
DeepMind built agents of increasing complexity:
- Agent A: a simple loop. The LLM (Gemini 3.1 Pro) generates an attempt, the Lean compiler checks it, the error goes back to the model, repeat.
- Agent D (full): subagents coordinated by an evolutionary algorithm with voting and Elo ratings, able to call AlphaProof, the RL-trained olympiad-level prover.
The most interesting caveat
In post-hoc analysis, Agent A, the simple loop, solved the same nine Erdős problems as the full agent, though at higher cost on the hardest ones. Josep Curto (UOC), speaking to Science Media Centre Spain, sees this as the main lesson: what moves the needle is a strong LLM with strict verifier feedback, not elaborate architecture. He also criticizes that costs count only successful runs and exclude AlphaProof inference.
Other caveats:
- Lean doesn't eliminate hallucination: it guarantees the proof holds for the statement written in Lean, not that the statement faithfully captures the real problem. The team had experts validate each statement.
- Non-random sample: the 353 are problems already formalized in Lean, out of a catalog of 1,200+.
- Mathematicians like Javier Aramayona (ICMAT) warn of results certified true that nobody actually understands.
Why it matters beyond math
The generate-and-verify pattern applies to any AI agent: the model proposes, an objective verifier (compiler, automated tests, schema validation, business rules) accepts or rejects. For businesses:
- Don't trust agent output without automated checks. Put tests, linters or validators in the loop.
- Start simple: a good model + strict feedback often beats complex architectures.
- Verify the spec: a verifier only guarantees what you asked for, so review the ask.
- Measure total cost, including failed attempts.
Sources
- Science, doi 10.1126/science.aej2213 (8/10/2026)
- arXiv 2605.22763: https://arxiv.org/abs/2605.22763
- GitHub, google-deepmind/alphaproof-nexus-results: https://github.com/google-deepmind/alphaproof-nexus-results
- Science Media Centre Spain: https://www.sciencemediacentre.es/en/alphaproof-nexus-new-ai-tool-designed-tackle-mathematical-proofs
- postcutoff: https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/
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