AI Claims 9 Erdős Problems: Google DeepMind’s AlphaProof Nexus Solves Decades-Old Math Puzzles
Google DeepMind's AlphaProof Nexus, powered by Gemini 3.1 Pro and the Lean theorem prover, has cracked 9 open problems from the Erdős list, including one unsolved for 56 years. It also proved 44 OEIS conjectures, solved a 15-year-old algebraic geometry problem, and improved a convex optimization bound — all at a cost of a few hundred dollars per problem.
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Key points
- AlphaProof Nexus solved 9 Erdős problems, 44 OEIS conjectures, and a 15-year-old algebraic geometry problem.
- The system uses a loop of LLM (Gemini 3.1 Pro) and Lean compiler feedback, with four increasingly sophisticated agent variants.
- Even the simplest agent (LLM + compiler) solved all 9 problems, emphasizing the power of verification feedback.
- Cost per problem is a few hundred dollars, demonstrating efficient AI-driven mathematical discovery.
Why it matters
This matters because alphaProof Nexus solved 9 Erdős problems, 44 OEIS conjectures, and a 15-year-old algebraic geometry problem.
Technical impact
May affect model selection, inference cost, product capability, and evaluation benchmarks.
Google DeepMind has unveiled AlphaProof Nexus, a framework that leverages the Gemini 3.1 Pro large language model and the Lean theorem prover to automate mathematical theorem proving. In a new paper, the team reports that they have solved nine long-standing open problems from the list of Hungarian mathematician Paul Erdős, a feat that includes one problem that had resisted solution for 56 years.
The system employs a multi-agent architecture. In its simplest form (Agent A), it consists solely of a loop: Gemini 3.1 Pro generates Lean proof code, the Lean compiler checks it for errors, and the model iteratively fixes any issues. This direct approach alone was sufficient to solve all nine problems. More advanced agents incorporate AlphaProof (a reinforcement learning-based tool), evolutionary algorithms for proof search, and combinations thereof, but the researchers note that the simpler methods were surprisingly effective — even outperforming multi-agent systems in some cases.
The solved Erdős problems span number theory, combinatorics, and geometry. For example, problem #12 (proposed in 1970) asks for an infinite set of integers with no element dividing the sum of two others, yet with positive natural density. The AI constructed such a set using the Chinese remainder theorem. Problem #125 (from 1996) deals with the additive complexity of numbers with restricted digit representations; the AI proved that the resulting set has zero density. Problem #846 (1992) concerns a point set in the plane that is local-globally inseparable, solved via Ramsey theory.
Beyond the Erdős problems, AlphaProof Nexus proved 44 open conjectures from the OEIS integer sequence encyclopedia, resolved a 15-year-old problem on the log-concavity of Hilbert functions in algebraic geometry, and improved a theoretical bound in convex optimization for the anchor gradient descent method.
The work highlights a growing trend: AI-powered theorem proving is becoming not only feasible but cost-effective. The entire set of proofs is open-sourced on GitHub, and the cost per problem was on the order of a few hundred dollars. As noted by Fields Medalist Terence Tao, the success rate of around 2% (9 out of 353 attempted Erdős problems) aligns with expectations for AI in this domain.
The implications are profound. The simplest agent — just a large language model guided by a strict verifier — was able to match more complex ensembles, suggesting that as LLMs improve, the need for elaborate tool combinations may diminish. For now, the era of AI mathematicians is firmly underway.