Frontiers
Quantities mathematicians have pushed for decades, drawn as the staircases they are. Every step is a paper or an announcement, cited from its source, and the steps where AI took part are marked.
9 frontiers65 steps cited12 moved with AI8 held with AI today
Every frontier
Bounded gaps between primes
Best known 186, GPT-6 Astra (OpenAI)· last moved 3 Sep 2026
8 steps since 2013· 3 moved with AI· lower is better
70,000,000, Yitang Zhang, 2013-05 4,680, Polymath8a, 2013-07 600, James Maynard, 2013-11-19 246, Polymath8b, 2014 240, Julia Stadlmann, 2026-08-31 236, Shiva Kintali, with Claude, ChatGPT, Codex and open-weight models, 2026-09-01 212, François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Ashvin Swaminathan, Jesse Thorner and Yunzhou Xie (Axiom Math), with AxiomProver, 2026-09-03 186, GPT-6 Astra (OpenAI), 2026-09-03
at least 3, G. Billing, 1938 at least 4, A. Wiman, 1945 at least 6, David Penney and Carl Pomerance, 1974 at least 7, David Penney and Carl Pomerance, 1975 at least 8, Fritz Grunewald and Rainer Zimmert, 1977 at least 9, Armand Brumer and Kenneth Kramer, 1977 at least 12, Jean-François Mestre, 1982 at least 14, Jean-François Mestre, 1986 at least 15, Jean-François Mestre, 1992 at least 17, Kazuhiko Nagao, 1992 at least 19, Stéphane Fermigier, 1992 at least 20, Kazuhiko Nagao, 1993 at least 21, Kazuhiko Nagao and Tomonori Kouya, 1994 at least 22, Stéphane Fermigier, 1997 at least 23, Robert Martin and William McMillen, 1998 at least 24, Robert Martin and William McMillen, 2000 at least 28, Noam Elkies, 2006 at least 29, Noam Elkies and Zev Klagsbrun, 2024 at least 30, Levent Alpöge and Ava Howell, with Claude, 2026-08-20 at least 31, Claude, Levent Alpöge and Ava Howell, 2026-08-23
Exponent of matrix multiplication
Best known 2.371177, Emilien Dupont and nine co-authors, with AlphaEvolve· last moved 17 Aug 2026
15 steps since 1969· 1 moved with AI· lower is better
2.8074, Volker Strassen, 1969 2.796, Victor Pan, 1978 2.7799, Dario Bini, Milvio Capovani, Grazia Lotti and Francesco Romani, 1979 2.522, Arnold Schönhage, 1981 2.517, Francesco Romani, 1982 2.496, Don Coppersmith and Shmuel Winograd, 1982 2.479, Volker Strassen, 1986 2.3755, Don Coppersmith and Shmuel Winograd, 1990 2.3729, Virginia Vassilevska Williams, 2012 2.3728639, François Le Gall, 2014-01 2.3728596, Josh Alman and Virginia Vassilevska Williams, 2020-10 2.371866, Ran Duan, Hongxun Wu and Renfei Zhou, 2022-11 2.371552, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Renfei Zhou, 2023-07 2.371339, Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Renfei Zhou, 2024-04 2.371177, Emilien Dupont and nine co-authors, with AlphaEvolve, 2026-08-17
more than 1/3, Norman Levinson, 1974 more than 2/5, J. Brian Conrey, 1989 more than 41%, Hung Bui, Brian Conrey and Matthew Young, 2010-02-22 more than 5/12, Kyle Pratt, Nicolas Robles, Alexandru Zaharescu and Dirk Zeindler, 2018-02-28 67.25%, Claude, with Levent Alpöge and Ralph Furman, 2026-08-10
566, John Leech and Neil Sloane, 1971 582, Victor Zinoviev and Thomas Ericson, 1999 592, Mikhail Ganzhinov, 2022-07-17 593, AlphaEvolve, 2025-06-16 604, Agents on EinsteinArena, 2026-06-09
1325, Jeff Kahn and Gil Kalai, 1993 298, Aicke Hinrichs and Christian Richter, 2003 65, Andriy Bondarenko, 2013-05 64, Thomas Jenrich, 2013-08 63, Max Grinsztajn, with GPT-5.5 Pro, 2026-05-26 63, Yibo Ji, with GPT 5.6 Sol, 2026-08-12, did not move the record
496 points, Yves Edel, 2004 512 points, FunSearch, 2023-12-14
49 multiplications, Strassen, applied recursively, 1969 47 multiplications, AlphaTensor, 2022-10-05
7 multiplications, Strassen, 1969 7 multiplications, AlphaTensor, 2022-10-05, did not move the record
How a frontier is drawn
The line is the best value known at each date, and it moves only when a step beats everything before it. Filled marks moved it, in ink for people and in clay where AI took part; grey marks did not; a hollow mark is a candidate, claimed and not accepted. Values that span a hundredfold or more are drawn on a logarithmic scale, and marks that would overlap are set side by side in their order. Each step names who reached it and cites where it appeared; older steps cite the paper, or a table that cites it, and the page says which.