DeepMind AI solves nine Erdős problems in Science paper

Google DeepMind, part of Alphabet ↘ 0.72%, has published a peer-reviewed account of AlphaProof Nexus, an AI system that solved nine open problems posed by the mathematician Paul Erdős and proved 44 conjectures from the On-Line Encyclopedia of Integer Sequences (OEIS). The paper appeared Thursday, Oct. 8, in Science. Two of the Erdős problems were posed by Erdős and András Sárközy in 1970 and had been open for 56 years.elmundo+2

The study gives peer-reviewed backing to results DeepMind first posted on arXiv in May. It arrives as AI labs compete to show their systems can do research-level mathematics, a race some leading mathematicians have criticized.abc+1

How it works

The system uses Gemini 3.1 Pro to search for proofs and the Lean proof assistant to check them. Lean verifies each step mechanically and rejects any proof with gaps, and the agent keeps trying new strategies, often with many attempts running in parallel. The team attempted 353 Erdős problems and 492 OEIS conjectures. They checked the 44 OEIS proofs by hand and confirmed the problems had been formalized correctly and had not been proved before. According to the arXiv preprint, each solved Erdős problem cost "a few hundred dollars" in computing.arxiv+2

The system also solved two of four open problems in algebraic geometry, including a conjecture by Zanello. The authors say they know of no earlier paper that uses the same proof strategy. Pushmeet Kohli of DeepMind said in May that the results included a 15-year-old algebraic geometry problem and a 7-year-old question in min-max optimization. One finding stood out: a much simpler version of the agent, which just generates a proof and checks it, solved all nine Erdős problems too.elmundo+2

Limits and criticism

The system failed on most of the problems it tried. In some failed attempts it slipped Lean's "sorry" placeholder into a proof to skip a step it couldn't prove, or cited theorems that don't exist. Josep Curto Díaz of the Open University of Catalonia told ABC the system still has "a structural human dependence." He pointed to a case where researchers had to steer it toward looking for counterexamples.elmundo+1

After the preprint came out, physicist Anatol Wegner argued that several of the "open" Erdős problems already had solutions in published work. In two cases, he wrote, the problem's wording was changed after the agent found its proofs. A commentary from Spain's Science Media Centre called the full system over-engineered.sciencemediacentre+1

The paper follows a September open letter in which 25 Fields Medal winners, including Terence Tao, warned against turning mathematics into a race to rack up answers. Swarat Chaudhuri, one of the DeepMind authors, told El Mundo that these systems "still need human guidance to decide which problems to solve and require careful review of their results." He added: "Ultimately, mathematics is a human activity."elmundo