Google DeepMind's AlphaProof Nexus Solves Decades-Old Math Problems for Just a Few Hundred Dollars

The AlphaProof Nexus framework autonomously solved 9 out of 353 unsolved Erdős problems. This included 2 problems that had remained unsolved for 56 years, as well as 44 conjectures from OEIS (Online Encyclopedia of Integer Sequences) and a 15-year-old problem related to the Hilbert function. The total inference cost was only a few hundred dollars per problem.

The system uses Gemini 3.1 Pro to generate proof steps in Lean, a formal language used for mathematical verification. The Lean compiler checks each step, and a loop that feeds error messages back to the model compensates for logical errors. This serves as a safeguard that offsets the logical reasoning weakness that has long plagued LLMs.

The system consists of four agent models.

• Agent (A): The simplest loop model using Gemini 3.1 Pro and compiler feedback.
• Agent (B): Adds reinforcement-learning-based AlphaProof to fill in missing parts of proofs.
• Agent (C): Introduces an evolutionary component inspired by AlphaEvolve.
• Agent (D): The complete form combining all of the above features.

Analysis showed that the simplest Agent (A) alone was able to prove all 9 of the solved Erdős problems. This suggests that the combination of improved LLM performance and compiler feedback is powerful, and shows that the technology trend is shifting from specialized systems to simple agentic loops.