Mathematical discovery and exploration can be done at scale

Source abstract

AlphaEvolve, introduced in A. Novikov et al . [“AlphaEvolve: A coding agent for scientific and algorithmic discovery” (Tech. Rep., Google DeepMind, 2025).], is a generic evolutionary coding agent that combines large language models (LLMs) with automated evaluation in a loop that proposes, tests, and refines algorithmic solutions. Here we showcase AlphaEvolve as a tool for autonomously discovering novel mathematical constructions and making progress on long-standing open problems. On 67 problems spanning mathematical analysis, combinatorics, geometry, and number theory, the system rediscovered the best known solutions in most cases and improved on them in several. In some instances, it also generalized results for finitely many input values into a formula valid for all inputs. Furthermore, we combine this methodology with Deep Think [Google DeepMind, Advanced Version of Gemini with Deep Think Officially Achieves Gold-Medal Standard at the International Mathematical Olympiad (Google DeepMind Blog, 2025).] and AlphaProof [Google DeepMind, AI Achieves Silver-Medal Standard Solving International Mathematical Olympiad Problems (Google DeepMind Blog, 2024).] in a broader framework where the additional proof assistants and reasoning systems provide automated proof generation and further mathematical insights. These results demonstrate that LLM-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or improving the best known results, and point to new modes of interaction between mathematicians and AI systems. AlphaEvolve explores vast search spaces to solve complex optimization problems at scale, often with significantly reduced preparation and computation time.