The Last Holdout Falls: AI Agents Help Mathematicians Close the M23 Inverse Galois Problem

In May 2026, in a room near the top of Caltech’s tallest building, the American Institute of Mathematics asked a roomful of mathematicians an unusual question: which open problems are ripe for AI’s particular talent — sifting through possibilities at larger-than-human scale? Rachel Pries of Colorado State University took the podium and nominated a candidate that “had been open for a very, very long time.” Within three months, that problem was dead.

The result, reported in a September 22 Scientific American feature and posted as the arXiv preprint The Mathieu group M₂₃ is a Galois group over ℚ (arXiv:2608.08538, submitted August 9), closes the final chapter of a 40-year saga in one of mathematics’ oldest unsolved questions — and it offers one of the cleanest demonstrations yet of what AI collaboration actually looks like at the frontier of pure mathematics. Not a model replacing mathematicians, and not a gimmick: agents doing combinatorial grunt work humans couldn’t feasibly do, while humans made every judgment call that mattered.

The problem: symmetry in reverse

The inverse Galois problem runs in the opposite direction from everything mathematicians learned in school about polynomial equations. For any polynomial with rational coefficients, algorithms can compute its Galois group — the catalog of ways its solutions can be shuffled while all algebraic truths stay intact. Galois theory, built by Évariste Galois in the early 19th century before his death in a duel at age 20, classifies polynomials by these hidden symmetry groups.

The forward direction is mechanical. The reverse question is not: given a specific finite group, does there exist a polynomial whose Galois group is exactly that group? Nobody knows the answer in general.

Most finite groups come in tidy infinite families — shuffling three roots, four roots, five, up an endless ladder. But the classification of finite simple groups revealed 26 rebels: the sporadic groups, which belong to no family and follow no pattern. Between 1984 and 1989, mathematicians realized 25 of the 26 as Galois groups over the rationals. One refused to cooperate: M₂₃, a Mathieu group acting on 23 points, discovered in the 19th century and stubborn ever since.

“There’s this one last holdout,” MIT’s Bjorn Poonen told Scientific American. “I think some people were even wondering whether there might be no polynomial giving M₂₃.”

The team and the method

After the Caltech workshop, participants bid on problems. The inverse Galois problem attracted so much interest that it assembled a six-person team of strangers: Pries and Poonen, joined by Xiaoyu Huang (Temple University), Blake Jackson (Institute for Computer-Aided Reasoning in Mathematics), Kyu-Hwan Lee (University of Connecticut), and Shaowu Zhang, a Caltech Ph.D. student.

Their strategy, laid out in the paper, was to find a non-rigid triple of conjugacy classes inside M₂₃ and compute Belyi maps to build an explicit regular Galois extension of ℚ(t) with Galois group M₂₃ — from which a degree-23 polynomial over the rationals falls out. The computational engine was the numerical Belyi map algorithm developed by Klug, Musty, Schiavone, Sijsling, and Voight, building on ideas of Hejhal and Stark.

That’s where the machines came in.

Ninety digits of nothing, then a miracle

The team set AI to work combing through M₂₃’s symmetry combinations, looking for workable surfaces. The search produced seven candidate surfaces, which they asked the AI to approximate numerically — hoping the decimals would crystallize into recognizable exact numbers. They didn’t. The AI kept requesting more precision, and by 90 digits the computation had exhausted the machine’s memory. The project stalled.

Then Zhang posted a suspicion in the team chat: the coordinates were wrong. The fix wasn’t subtle cleverness — it was another brute-force campaign, this time across new, more “harmonious” coordinate systems. The team describes what happened next as “miraculous” in their paper: they deployed a raft of AI agents on the re-coordinatized problem. Most failed. One came back excited about a single one of the seven surfaces, reporting — in the agent’s own words — “This might work!”

It did. A few more steps converted the numerical approximation of that surface into an explicit equation, and ultimately into a family of degree-23 polynomials — 23 roots to shuffle, symmetries forming exactly M₂₃. One representative from the paper begins x²³ − 184x²¹ − 1150x²⁰ + 26151x¹⁹ + ⋯, with integer coefficients running to trillions before the constant term closes at −3,150,159,884,154.

The clock is the striking part. Workshop in May, preprint in August. “We could do it very efficiently,” Kyu-Hwan Lee said. “That wasn’t really possible five years ago.”

Why it matters

The M₂₃ result is a small-but-mighty piece of an enormous puzzle. Mathematicians suspect a single master object — the absolute Galois group — encodes the symmetries of all polynomials at once, and mapping its structure is among number theory’s grandest ambitions. Every finite group realized as a Galois group over ℚ is a confirmed fragment of that map. The last sporadic fragment is now in place.

The episode also lands amid a crowded season of “AI does math” headlines — and it’s a useful calibration point. In an adjacent development, the Foundation for Science and AI Research (SAIR), whose founders include Fields Medalist Terence Tao, ran a crowdsourced competition to find polynomials for every group acting on 24 roots. The first phase concluded in late August with all 25,000 group-polynomial relationships realized. Despite AI being allowed and heavily used throughout, the winning team — two German mathematicians — used it for exactly one thing: writing their upload script.

“It was open to AI, and it was still these folks who did the best,” said Mount Holyoke’s Jen Paulhus, one of the competition’s organizers. “As a working mathematician, that heartened me.”

The lesson from M₂₃ is more nuanced than either hype or backlash. The agents didn’t conceive the strategy, choose the conjugacy classes, recognize that coordinates were the problem, or certify the final proof — humans did all of that. What the agents did was search spaces of symmetry combinations and coordinate systems at a breadth no human team could afford, and one of them flagged the needle. The frontier, it turns out, is a division of labor: taste, judgment, and proof remain human; the combinatorial ocean does not.