Google DeepMind AI Framework Cracks Nine Erdős Problems, Including Two Unsolved for 50 Years

Google DeepMind has developed and tested AlphaProof Nexus, an AI framework that autonomously searches for mathematical proofs and verifies reasoning using computer programs such as Lean. The findings were published in the latest issue of Science.

During testing, the system solved 9 previously unsolved problems named after mathematician Paul Erdős out of 353 attempted, including two that had remained open for more than 50 years. It also resolved 44 of 492 open conjectures in the Online Encyclopedia of Integer Sequences, and tackled research-level problems in areas including algebraic geometry, optimization, quantum optics and graph theory.

AlphaProof Nexus adjusts its proofs based on feedback from the Lean compiler. A more advanced version uses evolutionary algorithms to coordinate multiple sub-agents and calls on DeepMind's AlphaProof as a dedicated proof tool. The team found that a basic agent version also solved the 9 Erdős problems, though it required more computational resources on harder questions; that version primarily relied on commercial large language models and the Lean proof assistant.

Some mathematicians worry that large-scale AI use could reduce incentives to explore mathematical problems and understand proofs, while the team argues that even unsuccessful AI attempts may help mathematicians better understand problems and uncover new approaches.