Google Deepmind’s AlphaProof Nexus has produced proofs for nine open Erdős problems, two of which had remained unsolved for 56 years. The results, described in a study in Science, also include conjectures from an integer sequence database and problems in other areas of mathematics.
What the system solved
The Erdős problems are a collection of open mathematical questions. AlphaProof Nexus cracked nine of 353 such problems. The team shared those solutions online ahead of the paper, according to the report.
The system’s reported work extends beyond that set. It proved 44 conjectures from the OEIS integer sequence database, solved an open question in algebraic geometry, and improved a known bound in min-max optimization. The source describes min-max optimization as a technique used to train some AI models.
Together, these results cover several different kinds of mathematical questions. The report does not describe them as a single benchmark or claim that they settle every problem in those fields. Rather, it presents a collection of proofs and results across distinct areas.
How AlphaProof Nexus builds proofs
At the center of AlphaProof Nexus is Gemini 3.1 Pro, which drafts and revises proofs using Lean, a formal language for expressing mathematics. Lean checks each draft after every edit. When a check returns an error, that feedback guides the system’s next attempt.
This checking step matters because a proof must establish its conclusion through valid mathematical steps. The system can revise its work in response to Lean’s feedback, creating a cycle of drafting, checking and correction rather than relying on a single unchecked answer.
For harder sub-problems, AlphaProof Nexus can call on AlphaProof, a specialized proof tool. Google Deepmind reports that the system solved the nine Erdős problems even without AlphaProof. That detail distinguishes those solutions from results that depended on the specialized tool.
Peer review enters the AI math debate
The study arrives as researchers and organizations debate how AI-generated mathematics should be shared and evaluated. The report contrasts Deepmind’s peer-reviewed study with OpenAI’s recent release of hundreds of AI proofs on GitHub instead of submitting them to journals.
The Association for Human Mathematics, led by Terence Tao, called for a boycott. Against that backdrop, submitting the AlphaProof Nexus work for peer review is a notable part of how Deepmind presented its results.
Peer review gives the work a different route into the mathematical record than an online release alone. The reported results are still specific to the problems described in the study: nine Erdős problems, 44 OEIS conjectures, an open algebraic geometry question and an improved bound in min-max optimization.
What the results show
AlphaProof Nexus combines a general-purpose model, Gemini 3.1 Pro, with a formal proof-checking process in Lean and access to AlphaProof for harder sub-problems. The report’s account suggests that this setup can contribute to work on open questions, while Lean’s repeated checks provide a way to catch errors during proof development.
The results do not mean that every mathematical question is now within reach of AI. They do show that AlphaProof Nexus has produced proofs for a defined set of open problems and other conjectures, and that the team’s study has gone through peer review. How widely this approach can be applied remains an open question beyond the results described here.