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Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,10,6]]},"abstract":"<jats:p>\n                    AlphaEvolve, introduced in A. Novikov\n                    <jats:italic toggle=\"yes\">et al<\/jats:italic>\n                    . [\u201cAlphaEvolve: A coding agent for scientific and algorithmic discovery\u201d (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,\n                    <jats:italic toggle=\"yes\">Advanced Version of Gemini with Deep Think Officially Achieves Gold-Medal Standard at the International Mathematical Olympiad<\/jats:italic>\n                    (Google DeepMind Blog, 2025).] and AlphaProof [Google DeepMind,\n                    <jats:italic toggle=\"yes\">AI Achieves Silver-Medal Standard Solving International Mathematical Olympiad Problems<\/jats:italic>\n                    (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.\n                  <\/jats:p>","DOI":"10.1073\/pnas.2536158123","type":"journal-article","created":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T16:13:32Z","timestamp":1790784812000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Mathematical discovery and exploration can be done at scale"],"prefix":"10.1073","volume":"123","author":[{"given":"Bogdan","family":"Georgiev","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00971b260","id-type":"ROR","asserted-by":"publisher"}],"name":"Google DeepMind","place":["London, United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5962-0859","authenticated-orcid":false,"given":"Javier","family":"G\u00f3mez-Serrano","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/05gq02987","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Mathematics, Brown University","place":["Providence"]},{"name":"School of Mathematics, Institute for Advanced Study","place":["Princeton"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Terence","family":"Tao","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/046rm7j60","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Mathematics, University of California","place":["Los Angeles"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6423-3730","authenticated-orcid":false,"given":"Adam Zsolt","family":"Wagner","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00971b260","id-type":"ROR","asserted-by":"publisher"}],"name":"Google DeepMind","place":["London, United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,9,30]]},"reference":[{"key":"e_1_3_4_1_2","unstructured":"F. 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Hayat \u201cGlobal Lyapunov functions: A long-standing open problem in mathematics with symbolic transformers\u201d in Advances in Neural Information Processing Systems A. Globerson L. Mackey D. Belgrave A. Fan U. Paquet J. Tomczak C. Zhang Eds. (Curran Associates Inc. 2024) vol. 37.","DOI":"10.52202\/079017-2969"},{"key":"e_1_3_4_13_2","unstructured":"G. Swirszcz Advancing geometry with AI: Multi-agent generation of polytopes. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2502.05199 (Accessed 17 September 2026)."},{"key":"e_1_3_4_14_2","unstructured":"Y. Wang Discovery of unstable singularities. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2509.14185 (Accessed 17 September 2026)."},{"key":"e_1_3_4_15_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-023-06747-5"},{"key":"e_1_3_4_16_2","unstructured":"Google DeepMind AI Achieves Silver-Medal Standard Solving International Mathematical Olympiad Problems (Google DeepMind Blog 2024)."},{"key":"e_1_3_4_17_2","unstructured":"Google DeepMind Advanced Version of Gemini with Deep Think Officially Achieves Gold-Medal Standard at the International Mathematical Olympiad (Google DeepMind Blog 2025)."},{"key":"e_1_3_4_18_2","unstructured":"A. Wei Gold medal-level performance on the world\u2019s most prestigious math competition\u2014The International Math Olympiad (IMO) (2025). https:\/\/x.com\/alexwei_\/status\/1946477742855532918 (Accessed 17 September 2026)."},{"key":"e_1_3_4_19_2","unstructured":"J. S. Ellenberg et al. Generative modeling for mathematical discovery. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2503.11061 (Accessed 17 September 2026)."},{"key":"e_1_3_4_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-5060(08)70776-3"},{"key":"e_1_3_4_21_2","first-page":"17429","article-title":"Discovering symbolic models from deep learning with inductive biases","volume":"33","author":"Cranmer M.","year":"2020","unstructured":"M. Cranmer , Discovering symbolic models from deep learning with inductive biases. Adv. Neural Inf. Process. Syst. 33, 17429\u201317442 (2020).","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"e_1_3_4_22_2","unstructured":"G. Lample F. Charton \u201cDeep learning for symbolic mathematics\u201d in International Conference on Learning Representations (ICLR) (2020)."},{"key":"e_1_3_4_23_2","unstructured":"F. 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Yang Formal mathematical reasoning: A new frontier in AI. arXiv [Preprint] (2024). https:\/\/doi.org\/10.48550\/arXiv.2412.16075 (Accessed 17 September 2026)."},{"key":"e_1_3_4_29_2","doi-asserted-by":"crossref","unstructured":"J. Goedgebeur J. Jooken G. Joret T. Van den Eede Improved lower bounds on the maximum size of graphs with girth 5. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2508.05562 (Accessed 17 September 2026).","DOI":"10.1080\/10586458.2026.2731953"},{"key":"e_1_3_4_30_2","unstructured":"T. Tao New Nikodym set constructions over finite fields. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2511.07721 (Accessed 17 September 2026)."},{"key":"e_1_3_4_31_2","unstructured":"T. Tao Sum-difference exponents for boundedly many slopes and rational complexity. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2511.15135 (Accessed 17 September 2026)."},{"key":"e_1_3_4_32_2","unstructured":"A. Nagda P. Raghavan A. Thakurta Reinforced generation of combinatorial structures: Applications to complexity theory. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2509.18057 (Accessed 17 September 2026)."},{"key":"e_1_3_4_33_2","unstructured":"A. Sharma OpenEvolve: An open-source evolutionary coding agent (Open-source implementation of AlphaEvolve) (2025). https:\/\/github.com\/codelion\/openevolve (Accessed 17 September 2026)."},{"key":"e_1_3_4_34_2","unstructured":"R. T. Lange ShinkaEvolve: Towards open-ended and sample-efficient program evolution. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2509.19349 (Accessed 17 September 2026)."},{"key":"e_1_3_4_35_2","unstructured":"G. Liu Y. Zhu J. Chen M. Jiang Scientific algorithm discovery by augmenting AlphaEvolve with Deep Research. arXiv [Preprint] (2025). https:\/\/doi.org\/10.48550\/arXiv.2510.06056 (Accessed 17 September 2026)."}],"container-title":["Proceedings of the National Academy of Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.2536158123","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.2536158123","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T16:13:53Z","timestamp":1790784833000},"score":1,"resource":{"primary":{"URL":"https:\/\/pnas.org\/doi\/10.1073\/pnas.2536158123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,30]]},"references-count":35,"journal-issue":{"issue":"40","published-print":{"date-parts":[[2026,10,6]]}},"alternative-id":["10.1073\/pnas.2536158123"],"URL":"https:\/\/doi.org\/10.1073\/pnas.2536158123","relation":{},"ISSN":["0027-8424","1091-6490"],"issn-type":[{"value":"0027-8424","type":"print"},{"value":"1091-6490","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,30]]},"assertion":[{"value":"2025-12-12","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-07-22","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-09-30","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e2536158123"}}