Google DeepMind’s AlphaProof Nexus solved nine of 353 open Erdős problems and proved 44 conjectures from the Online Encyclopedia of Integer Sequences, according to research published in Science. Two of the Erdős problems had remained open for 56 years.
The system uses Gemini 3.1 Pro to draft and revise proofs in Lean, a formal language whose checker can verify every logical step. Errors from Lean guide the model’s next attempt. For difficult subproblems, Nexus can call AlphaProof, a specialized proof system, although DeepMind says the nine Erdős results did not require that tool.
The work also addressed an open algebraic-geometry question and improved a known bound in min-max optimization. Formal verification matters because a convincing mathematical explanation can still contain a hidden logical gap; Lean checks whether the proof follows from its declared assumptions.
Peer review distinguishes this release from large batches of AI-generated proofs posted without conventional review. It does not mean every output of the system is correct or that mathematicians are no longer needed. Researchers must select problems, inspect assumptions and decide whether a machine-verified result is mathematically meaningful. The result shows that model-guided formal proof can now contribute to open problems, not merely classroom exercises.