Recently, the integration of artificial intelligence (AI) into the realm of mathematics has captured considerable interest. OpenAI made headlines by releasing a batch of 722 mathematical proofs, all pending verification, a move that stirred up debate within the mathematical community. Simultaneously, Google DeepMind has achieved notable advancements in AI-driven mathematical exploration, with its findings published in the prestigious journal Science and earning recognition from the academic world.

Google DeepMind introduced the AI mathematical proof agent framework, AlphaProof Nexus, which is underpinned by the power of Gemini. This innovative framework successfully tackled nine open problems posed by the renowned mathematician Paul Erdős, including two that had baffled experts for 56 years. Furthermore, it substantiated 44 conjectures listed in the OEIS Encyclopedia of Integer Sequences, addressed an algebraic geometry problem that had lingered unresolved for 15 years, and refined long-standing theoretical limits in the field of convex optimization.

The central workflow of AlphaProof Nexus involves Gemini generating proofs, which are then scrutinized by Lean—a proof assistant software—for verification. Lean also provides feedback on any errors detected, prompting iterative refinements until the proof is deemed valid. The framework encompasses four agents, each configured differently to optimize performance. Controlled experiments have revealed that even the simplest configuration is capable of solving all nine Erdős problems. When it comes to more challenging problems, the full version of the framework proves to be 2 to 5 times more cost-effective than the basic version, with the cost per problem hovering around several hundred dollars.

Currently, the complete set of Lean proof codes developed by the AlphaProof Nexus framework has been made accessible to the public, fostering further research and collaboration in the mathematical community.