DeepMind's AlphaProof Nexus Solves 9 Open Erdős Problems by Pairing Gemini With the Lean Proof Checker

Overview

A team at Google DeepMind has reported an artificial-intelligence system that autonomously resolved nine previously open problems from the Erdős problem catalog by pairing large language models with the Lean proof assistant, which mechanically checks each logical step. According to the team’s preprint, Advancing Mathematics Research with AI-Driven Formal Proof Search, the framework — named AlphaProof Nexus — “autonomously solved 9 Erdős problems out of 353 attempted” and “proved 44/492 OEIS conjectures” drawn from the Online Encyclopedia of Integer Sequences. The work was first posted to arXiv on May 21, 2026, as reported by Tech Times.

The result is notable less for any single proof than for the method: rather than asking a model to produce a convincing natural-language argument, the system forces every candidate proof through a formal compiler that accepts only fully verified logic. That design directly targets the unreliability that has limited language models as research tools.

What We Know

The headline figures are consistent across the preprint and secondary coverage. WinBuzzer reported that “AlphaProof Nexus was credited with resolving 9 of 353 open Erdős problems,” and Tech Times wrote that the system “solved nine, including two that had been open for 56 years, and proved 44 conjectures from the Online Encyclopedia of Integer Sequences.”

At the center of the approach is Lean, described by Tech Times as “a formal proof assistant that checks every logical step in a mathematical argument against a set of axioms.” Because Lean only certifies an argument when every step type-checks, a language model cannot bluff its way to a passing proof — the compiler either confirms the chain of reasoning or rejects it. WinBuzzer summarized researchers’ framing of the design as “the power of compiler feedback in grounding LLM reasoning.”

The preprint specifies the models behind the agent. The prover subagents are based on Gemini 3.1 Pro, while a less expensive Gemini 3.0 Flash handles rating, according to the arXiv paper. WinBuzzer likewise noted that “AlphaProof Nexus pairs Gemini 3.1 Pro with Lean.”

The full system is more than a single model in a loop. The preprint describes a “full-featured agent” whose prover subagents are coordinated by “an evolutionary algorithm,” using a “Population Database,” “Elo ratings,” and a “P-UCB formula” to decide which partial proof sketches to expand. The agent can also call DeepMind’s earlier olympiad-level prover, AlphaProof, as a focused tool: per the paper, it “can query AlphaProof to fill out missing parts of sketches,” running in “tree search inference mode” and returning a “proof, disproof, or failure message.”

The specific Erdős problems the full-featured agent solved are listed in the paper as #12(i), #12(ii), #125, #138, #152, #741(i), #741(ii), #846, and a variant of #26, according to the arXiv paper.

Cost is one of the more striking claims. The preprint reports a “per-problem cost of a few hundred dollars,” a figure echoed by both WinBuzzer and Tech Times, which wrote that “inference costs ran to a few hundred dollars per problem.”

What We Don’t Know

The authors are explicit that the win rate is low and uneven. The full-featured agent cleared 9 of 353 Erdős problems and 44 of 492 OEIS conjectures — solving the large majority of neither set. The preprint cautions that “even most Erdős problems remain out of reach,” and that the system’s “successes are concentrated in areas such as combinatorics, convex optimization, and number theory, where Lean’s mathematics library is mature.” In other words, the method leans heavily on the prior human effort encoded in Lean’s library, and falters in fields where that formal scaffolding is thinner.

The preprint is also a non-peer-reviewed v1 at the time of the reporting above, later revised. The proofs themselves are machine-checked by Lean, which is a meaningfully stronger guarantee than human refereeing for the logical validity of each argument; what remains open is the broader significance of the particular problems solved and how the results will be received once mathematicians scrutinize the formalizations.

There is also a question of framing. WinBuzzer reported that DeepMind chief executive Demis Hassabis “said the system is ‘still not AGI,’” and Tech Times noted that DeepMind’s leadership “was careful to note the system is ‘still not AGI.’” The hedge is a reminder that solving curated open problems with a mature formal library is not the same as open-ended mathematical research.

Analysis

The pattern on display is the same one that has reshaped competitive theorem-proving over the past two years: a generative model proposes, and a verifier disposes. What makes formal mathematics an unusually clean testbed for AI is that the verifier is not another fallible model but a compiler with a fixed set of axioms. A proof that Lean accepts is correct in a way that a graded essay or a benchmark answer is not, which removes the usual worry that a model is merely producing plausible-sounding text.

That property is why the cost figure matters as much as the solve count. If a research-grade result can be reached for a few hundred dollars of inference and then independently certified by a compiler, the economics of attacking a long list of open conjectures change. The bottleneck shifts from whether a model can be trusted to how many problems can be formalized in Lean in the first place — which is exactly where the paper says its successes cluster.

The limitation the authors flag is therefore the most informative part of the work. Concentrating wins in domains with a mature Lean library suggests that, for now, these systems extend human formalization effort rather than replacing it. The frontier they are pushing is not abstract reasoning in a vacuum but the slow, collective project of encoding mathematics in a form a machine can check.