Google Deepmind has developed a new AI framework called AlphaProof Nexus that can autonomously solve decades-old mathematical problems at a fraction of the cost of traditional methods.
The system cracked nine out of 353 open Erdős problems it attempted, including two questions that had remained unanswered for 56 years. It also proved 44 out of 492 open conjectures from the Online Encyclopedia of Integer Sequences (OEIS), settled a 15-year-old question about Hilbert functions in algebraic geometry, and improved a known bound in convex optimization. According to the research paper, inference costs ran just a few hundred dollars per problem.
How AlphaProof Nexus works
Unlike pure natural-language approaches such as OpenAI's recent solution, AlphaProof Nexus does not rely on the language model to carry the entire logical chain on its own. The underlying language model, Gemini 3.1 Pro, generates proof steps in Lean's formal programming language. The Lean compiler then checks each step, and error messages feed back into the next attempt.
This approach grounds the language model through symbolic feedback, offsetting the known weaknesses of large language models in logical reasoning. Humans only step in at the very end to verify the results.
The system consists of four agent variants with increasing complexity. The simplest variant, Agent (A), uses independent sub-agents running Gemini 3.1 Pro in loops. The language model generates proof steps, the Lean compiler checks them, and error messages guide the next attempt.
Agent (B) adds queries to AlphaProof, Google's reinforcement-learning system for olympiad math, which can fill in missing proof segments. Agent (C) introduces an evolutionary component inspired by AlphaEvolve, where sub-agents share a common population of proof sketches. Rating agents built on Gemini 3.0 Flash score these sketches for plausibility and novelty, then rank them using an Elo system. The fully equipped Agent (D) combines all of these capabilities.
Surprising results from the simplest agent
Agent (D) was used for the Erdős problems. However, a post-hoc analysis revealed a surprise: the simplest Agent (A), which only uses an LLM and compiler feedback, could also prove all nine solved Erdős problems, though it was more expensive on the hardest ones.
The researchers attribute this success to two factors: rapid improvement in the underlying language models and the "power of compiler feedback in grounding LLM reasoning." The fully equipped agent still holds an edge on the toughest tasks, but that lead could shrink as LLMs improve. The researchers describe "an ongoing shift from specialized trained systems toward simple agentic loops as LLMs become more capable."
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The system's successes cluster in areas like combinatorics, convex optimization, and number theory, where Lean's math library Mathlib is mature and problems break down into manageable sub-goals. Most Erdős problems remained out of reach, "let alone problems that require extensive new theory," the researchers write. The agents also inherit the unreliability of the underlying language models.
Value beyond solved problems
Mathematicians who worked with the system reported that even failed proof attempts deepened their understanding of a problem. The researchers noted that "AI-driven formal proof search can serve not only to solve problems but to deepen human understanding."
Because the sketches were formal, experts could focus on unsolved sub-goals instead of re-checking the entire argument from scratch. The agents also proved effective at catching flawed formalizations in the literature. "Formal verification can serve as a filter for determining which proofs merit human review," the authors wrote.
The system is already being used in ongoing research on quantum optics and graph theory, according to the paper. All Lean proofs and selected natural-language proofs are available on GitHub.
Context in the AI math landscape
OpenAI recently used a proprietary reasoning model to disprove Erdős's unit-distance conjecture. Fields Medalist Tim Gowers called it "a milestone in AI mathematics." Before that, GPT-5.2 Pro helped solve Erdős problem #281, with Terence Tao calling the case "perhaps the most unambiguous instance" of an LLM solving an open math problem. Thereafter, GPT-5.4 solved another Erdős problem.
In some ways, those results are more impressive than Deepmind's approach, as the language model had to carry the entire logical chain through natural language without a Lean compiler checking each step. AlphaProof Nexus is more systematic and scalable, but tackles a different goal: building a reliable AI tool for everyday math research.
Tao previously warned against reading too much into the headlines. AI's actual success rate on Erdős problems sits at just one to two percent, concentrated on easier tasks. Google's system cracked only nine out of 353 problems, which aligns almost exactly with Tao's two-percent bar.
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