Mathematical exploration and discovery at scale

2025-11-03

AI Summary

This work addresses the challenge of solving complex mathematical optimization problems through human–AI collaboration. Method: We propose AlphaEvolve—a novel autonomous coding agent that synergistically integrates large language models (LLMs) with evolutionary algorithms. Operating via iterative “generate–evaluate–mutate–select” cycles, AlphaEvolve automatically discovers mathematical constructions, induces general formulas, and collaborates with DeepThink and AlphaProof for automated formal proof and deep deductive reasoning. Crucially, it unifies symbolic computation and formal verification within an LLM-guided evolutionary search framework. Contribution/Results: Evaluated on 67 open mathematical problems spanning diverse domains—including combinatorics, number theory, and discrete optimization—AlphaEvolve reproduces most known optimal solutions and achieves breakthrough results on several previously unresolved instances. The framework significantly reduces both manual intervention and computational overhead, establishing a new paradigm for AI–mathematician co-reasoning in advanced mathematical discovery.

Technology Category

• Search and Optimization: Evolutionary Computation
• Machine Learning: Evolutionary Learning
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

AlphaEvolve is a generic evolutionary coding agent that combines the generative capabilities of LLMs with automated evaluation in an iterative evolutionary framework that proposes, tests, and refines algorithmic solutions to challenging scientific and practical problems. In this paper we showcase AlphaEvolve as a tool for autonomously discovering novel mathematical constructions and advancing our understanding of long-standing open problems. To demonstrate its breadth, we considered a list of 67 problems spanning mathematical analysis, combinatorics, geometry, and number theory. The system rediscovered the best known solutions in most of the cases and discovered improved solutions in several. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. These results demonstrate that large language model-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or even improving the best known results, highlighting the potential for significant new ways of interaction between mathematicians and AI systems. We present AlphaEvolve as a powerful new tool for mathematical discovery, capable of exploring vast search spaces to solve complex optimization problems at scale, often with significantly reduced requirements on preparation and computation time.

Problem

• Autonomously discovers novel mathematical constructions for open problems
• Proposes and refines algorithmic solutions through evolutionary framework
• Explores vast search spaces to solve complex optimization problems

Innovation

• Evolutionary coding agent combining LLMs with automated evaluation
• Iterative framework proposing testing refining algorithmic solutions
• Integration with proof-assistants for automated proof generation insights