Mathematicians Use AI to Solve Last Open M23 Galois Case

A six-mathematician team has used AI agents to construct a degree-23 polynomial whose Galois group is M23, the sporadic group that had sat unsolved since the 1980s, Scientific American reports.

"There's this one last holdout," MIT's Bjorn Poonen told the magazine. "I think some people were even wondering whether there might be no polynomial giving M23." The other 25 sporadic groups had been matched to explicit polynomials decades ago; M23 was the empty seat at the table.

The team, led by Rachel Pries of Colorado State and including Caltech Ph.D. student Shaowu Zhang alongside collaborators at Connecticut, Temple, and the Institute for Computer-Aided Reasoning in Mathematics, ran AI agents to hunt for a geometric surface with the right symmetries, then converted a workable one into polynomial equations. Zhang identified a coordinate-system problem that had stalled earlier attempts; an agent, given fresh coordinates, produced the breakthrough surface and reported back, "This might work!" The team called the moment "miraculous" in their arXiv preprint. Three months, start to finish.

"We could do it very efficiently," said Kyu-Hwan Lee of the University of Connecticut. "That wasn't really possible five years ago."

Running in parallel, a crowdsourced contest from Terence Tao's SAIR foundation, organized by Jen Paulhus of Mount Holyoke, catalogued all 25,000 polynomial-to-Galois-group correspondences acting on 24 roots. The winning entry came from a German team whose only use of AI, per the article, was writing an upload script. "It was open to AI, and it was still these folks who did the best," Paulhus said. "As a working mathematician, that heartened me."