Mathematicians Say They've Solved the Last Holdout in a 40-Year-Old Puzzle, With AI Helping Along the Way

Six mathematicians today posted a paper that finds an ordinary equation hiding a rare symmetry pattern called M23, the last of 26 unusual patterns that had never been matched to such an equation. They write that they used AI chatbots throughout the research, but checked the final answer with standard math software and no AI.

Six mathematicians, Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries and Shaowu Zhang, today posted a paper online that says it closes out a search mathematicians have been working on for decades.

The idea behind it takes a moment. The solutions of an equation are tied together by hidden symmetries: ways you can shuffle the solutions around while keeping every relationship between them intact. Each equation has its own pattern of shuffles. Mathematicians have catalogued the basic building-block patterns, and 26 of them are oddballs that fit no family, called the "sporadic" ones. A long-standing question asks whether every such pattern shows up as the symmetry of some ordinary equation, one written with plain whole numbers and fractions.

For the sporadic patterns, researchers found matching equations for 25 of the 26 between 1984 and 1989. The last one, called M23, "remained unrealized until now, despite many attempts," the authors write. Their paper gives an explicit equation, with x raised to the 23rd power and very large whole-number coefficients, whose solutions have exactly the M23 pattern.

The authors are unusually specific about AI. They began the project at a May 2026 workshop titled "AI and number theory" at the American Institute of Mathematics. They write that "no text in this article was written by AI," but that they used three AI chatbots, Anthropic's Claude Fable 5 and Claude Opus 4.8 and OpenAI's ChatGPT 5.6 Sol, for searching past research, writing computer code, testing ideas, ruling out other approaches, planning the computations, checking results and proofreading. They thank Anthropic for giving each of them a three-month Claude subscription. And they draw a clear line at the end: "Our final results were verified in Magma and PARI/GP without the use of AI," two long-established math programs that do exact calculation.

The paper is a preprint, meaning it has been shared publicly but not yet checked by outside reviewers for a journal.