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Mathematical exploration and discovery at scale

Paper · 2025 · Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao et al.

Summary

The paper asks whether an LLM-driven agent can help with constructive mathematics: building explicit objects (graphs, packings, functions, point sets) that reach strong numerical bounds on open problems. The authors test AlphaEvolve, an evolutionary coding agent from Google DeepMind. In AlphaEvolve, a language model repeatedly rewrites programs, and an automatic evaluator scores each program and keeps the best ones.

AlphaEvolve does not evolve objects directly. Its standard "search mode" evolves search heuristics. Each heuristic gets a fixed compute budget to improve the current best construction. A separate "generalizer mode" asks for one program that works for every input size n, so the system can turn small cases into a general formula.

They ran it on 67 problems from analysis, combinatorics, geometry and number theory. It matched the known records in most cases and beat them in several. Some outputs were later proved rigorously with Deep Think and AlphaProof. The work suggests that many problems can be explored at once with little expert setup.

Key points

  • The benchmark has 67 problems. AlphaEvolve recovered the best known constructions for most of them and improved on several, but it did not succeed on all.
  • Compared with its predecessor FunSearch, AlphaEvolve evolves whole code files of up to hundreds of lines in any language. It needs only thousands of LLM samples rather than millions, and it gains from state-of-the-art LLMs.
  • In search mode, each evolved program is an "improver" with a time limit (for example 100 seconds). Chaining such heuristics produces specialized stages, first broad exploration and then fine-tuning.
  • In generalizer mode, a construction for the Nikodym problem led to a follow-up paper by Terence Tao (arXiv:2511.07721).
  • Setting up a problem usually took a few hours at most. Deep Think and AlphaProof were added to the pipeline for proofs and formalization, and the authors are releasing a public Repository of Problems with code.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Fields, topics and concepts

📚 Shelves 3The field of science this work belongs to (OpenAlex fields).Computer Science · 19,503 worksArtificial Intelligence · 18,805 worksComputational Theory and Mathematics · 496 works

🧭 Topics 1Research subjects and languages the work is about.Reasoning · 1,948 works

Cite

Cite as Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao and Adam Zsolt Wagner. 2025. Mathematical exploration and discovery at scale. arXiv:2511.02864.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.48550/arxiv.2511.02864 · arXiv 2511.02864

@misc{georgiev2025mathematical,
  title = {{Mathematical exploration and discovery at scale}},
  author = {Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao and Adam Zsolt Wagner},
  year = {2025},
  doi = {10.48550/arxiv.2511.02864},
  eprint = {2511.02864},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2511.02864}
}

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