Google DeepMind's AlphaProof Nexus autonomously solved nine open Erdős problems and 44 conjectures by combining language models with formal proof verification

AlphaProof Nexus solved 9 out of 353 open Erdős problems attempted, including two questions unanswered for 56 years, and proved 44 out of 492 open conjectures from the Online Encyclopedia of Integer Sequences (OEIS). The system also settled a 15-year-old question about Hilbert functions in algebraic geometry and improved a known bound in convex optimization. Inference costs ran just a few hundred dollars per problem.

The system uses Gemini 3.1 Pro to generate proof steps in Lean's formal language, then a compiler checks each step and feeds error messages back for refinement—grounding the language model in symbolic feedback rather than relying on natural language alone. Four agent variants exist with increasing complexity, from a simple loop of LLM generation and compiler feedback (Agent A) to a fully equipped version combining reinforcement learning, evolutionary ranking, and feedback systems (Agent D).

A surprising finding emerged: Agent (A), the simplest variant using only an LLM and compiler feedback, could also prove all nine solved Erdős problems, albeit at higher cost on the hardest ones. Researchers attribute this to rapid improvement in underlying language models and the 'power of compiler feedback in grounding LLM reasoning,' suggesting a broader shift 'from specialized trained systems toward simple agentic loops as LLMs become more capable.'