Mathematical exploration and discovery at scale – Terence Tao et al. (arxiv.org)

Researchers introduce AlphaEvolve, a generative, evolutionary coding agent that pairs large language models with automated evaluation in an iterative propose-test-refine loop to discover algorithmic and mathematical constructions at scale. Applied to a benchmark of 67 problems across analysis, combinatorics, geometry and number theory, the system rediscovered most best-known solutions, produced improvements on several problems, and in some cases generalized patterns observed on finite inputs into formulas valid for all inputs. AlphaEvolve’s pipeline uses LLMs to generate candidate code or constructions, an automated fitness/evaluation stage to test and score them, and evolutionary operators to iteratively refine promising candidates. Beyond raw discovery, the authors integrate AlphaEvolve with proof-assistant tools (Deep Think, AlphaProof) to move from empirical constructions to automated proof attempts, showing a path from conjecture generation to formal verification. The work demonstrates that LLM-guided evolutionary search can complement human intuition, explore vast combinatorial search spaces with relatively low pre-specialization, and accelerate mathematical discovery. Key implications include new workflows for mathematicians that combine generative models, automated testing and proof tools, plus questions about rigorous verification, reproducibility and computational costs as such systems scale.