Google DeepMind’s AlphaProof Nexus, an AI system that searches for mathematical proofs and has every step checked by the Lean proof language, solved 9 of the 353 open Erdős problems it attempted, according to a paper published in Science on 8 October 2026. Two of them had been open for 56 years, the authors write, and their most capable agent solved the problems at an inference cost of a few hundred dollars per problem.
The system also proved 44 of 492 open conjectures from the Online Encyclopedia of Integer Sequences (OEIS), the paper says. An earlier version appeared as an arXiv preprint on 21 May 2026; this is a catch-up on the peer-reviewed publication, which we had not covered when it came out.
Business Pill · VERIFICATION COST
A one-minute explainer of verification cost: what it takes to check that an output is right, and why that check often costs more than producing the output. It teaches the general idea only and says nothing about any organisation in this story.
The key insight: As we read it, the paper’s most useful finding is about the loop rather than the specialist: once Lean checks every step, a plain generate-and-check loop on a general model reached the same nine Erdős results, and the extra machinery mattered mostly on the hardest problems.
In the basic agent, the paper says, independent prover subagents based on Gemini 3.1 Pro reason through a problem and edit a proof sketch with a search-and-replace tool. After each turn the sketch is compiled in Lean, and if it does not compile, Lean’s error message is used to direct the next turn.
The full-featured agent adds more machinery, according to the paper. It coordinates groups of subagents through an evolutionary algorithm, uses rating agents based on the less expensive Gemini 3.0 Flash to rank unfinished sketches into Elo ratings, and sends subgoals to AlphaProof, DeepMind’s reinforcement-learning prover, as a specialized proof tool.
What the Google DeepMind team reports in its Science paper of 8 October 2026: 9 of 353 open Erdős problems solved, two of them open for 56 years; 44 of 492 open OEIS conjectures proved; a 15-year-old question on Hilbert functions in algebraic geometry resolved; an open bound in convex optimization improved; and all Lean proofs published on GitHub. The paper’s results.
What It Proved
Beyond the Erdős and OEIS results, the authors say the agent resolved a 15-year-old open question on Hilbert functions in algebraic geometry and improved an open bound in convex optimization by discovering a novel algorithmic parameter schedule. They say it also identified several misformalizations in the literature and helped resolve an open problem from Ben Green’s list.
For Erdős problems #125 and #741(i), the paper says, the word “density” in the original statements was amended after the agent found proofs that used a different meaning of it, and the agent still resolved the corrected questions. The authors say all Lean proofs are available in a public GitHub repository, google-deepmind/alphaproof-nexus-results.
The Simple Loop Did Almost as Well
The authors compared the full-featured agent with simpler versions on the 9 Erdős problems it solved. “Remarkably, the basic agent solved all 9 problems, though at a higher cost on the harder problems,” they write, while the full-featured agent did significantly better on some challenging problems such as Erdős #125.
AlphaProof on its own, in tree-search mode, and versions of the basic agent built on smaller models (Gemini 3.0 Flash and Gemini 3.1 Flash-Lite) “could not solve any of the problems,” the paper says. The authors write that their results “point to an ongoing shift from specialized trained systems toward simple agentic loops as LLMs become more capable.”
What It Costs
The paper puts the inference cost at a few hundred dollars per solved Erdős problem, and says AlphaProof itself cost approximately 27.5 TPU hours ($60) per problem on v6e TPUs. The authors add that they do not intend the dollar figures as a comparison of AI and human mathematical labor.
They also say the reported costs have high variance and do not capture the full cost of discovery: they ran the full-featured agent on all 353 Erdős problems, and identifying the tractable ones “was itself a significant computational investment.”
Why the Formal Check Matters
The paper describes failure modes that Lean caught. On several problems, it says, the top-rated sketches relied on lemmas the agent claimed were established results in the literature, and on manual inspection those lemmas proved to be hallucinations. “These failure modes underscore the value of end-to-end formal verification,” the authors write.
In a related Perspective in Science, Jeremy Avigad and Matthew Ballard write that the authors report “that even unsuccessful proof attempts by AI could help them understand the problems better and make progress on solving them,” according to the journal’s release.
As we read it, formal verification changes who has to trust the output. The paper says hallucinated lemmas showed up in the top-rated sketches; Lean, not a human reviewer, is what kept them out of the results.
As we read it, the comparison favors the general model in a loop: the paper says standalone AlphaProof in tree-search mode solved none of the nine problems, while the basic agent built on Gemini 3.1 Pro solved all of them.
As we read it, the dollar figures are a floor rather than a price list: the authors say the cost of searching all 353 problems to find the tractable ones is not in the per-problem number.
Tsoukalas et al., Science, 8 October 2026 (preprint text)
“point to an ongoing shift from specialized trained systems toward simple agentic loops as LLMs become more capable.”
Three Implications
MATHEMATICIANS The paper says the agent is aiding ongoing research in quantum optics and graph theory, and that it identified several misformalizations in the literature.
AI LABS The authors say a basic agent alternating LLM generation with Lean verification replicated the Erdős successes, at a higher cost on the hardest problems.
ANYONE PRICING AI WORK The paper puts the inference cost at a few hundred dollars per solved Erdős problem and AlphaProof at about 27.5 TPU hours ($60) per problem, and says these figures leave out the cost of finding tractable problems.
The framework puts it this way: “The surface, not the model, governs autonomy.”
As we read it, AlphaProof Nexus is a mathematical case of that claim: the paper credits its results to the loop around Gemini, the generate-check-revise cycle with Lean, more than to the specialized prover it was built to call.
Solve rates on the paper’s two systematic test sets: 44 of 492 open OEIS conjectures (about 8.9%) and 9 of 353 open Erdős problems (about 2.5%). The counts are the paper’s; the percentages are our arithmetic.
What Is Not Established
We read the journal’s news release and the authors’ May 2026 arXiv preprint in full; the Science article itself is paywalled, and the published version may differ from the preprint in details. Nine of 353 attempted Erdős problems is about 2.5% by our arithmetic, and the authors say their cost figures leave out the work of finding which problems were tractable. Neither text we read says whether the system will be offered outside research use, and we have not checked the proofs ourselves.
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Google DeepMind’s team says AlphaProof Nexus solved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures, with every proof checked by Lean. Its own comparison found that a basic generate-and-verify loop on Gemini 3.1 Pro solved the same 9 Erdős problems, at higher cost on the hardest ones.
A note on sourcing. On 13 October 2026 we read in full the AAAS/Science news release on the paper published in Science on 8 October 2026 and the authors’ arXiv preprint of 21 May 2026. The Science article itself is paywalled and was not read. Every result, figure and quote about the paper here is the authors’ or the journal’s, except the 2.5% and 8.9% solve rates, which are our arithmetic; the Lens quote is from our own Business Engineer framework. Nothing here is investment advice.
Gennaro Cuofano is a CRO and tech executive who has worked in AI since late 2015, starting with bringing NLP, natural-language generation, voice and chatbot products to the marketing industry. His background is in law and finance: he holds a Master's degree in Law and an International MBA with an emphasis on corporate finance (LUISS Business School and the University of San Diego, 2012), and worked as a financial analyst at a real-estate investment firm in San Diego and as an assistant controller. His work focuses on business model strategy, business engineering and, more broadly, structural analysis: how companies make money, read from their own filings. He created FourWeekMBA and leads research there and at The Business Engineer, his newsletter on AI and business strategy, with over 95,000 subscribers and more than 1,000 published analyses. His writing has also appeared on Entrepreneur, HackerNoon and Search Engine People. Find him on LinkedIn and Substack. See how we source.
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