Five years ago, mathematicians Dawei Chen and Quentin Gendron confronted a challenging area of algebraic geometry tied to differentials, crucial in measuring distances on curved surfaces. Their work led them to a significant hurdle; they needed to justify a peculiar formula from number theory but were unable to do so. Consequently, they presented their findings as a conjecture instead of a theorem.
Seeking resolution, Chen spent considerable time experimenting with ChatGPT for potential solutions, but without success. Fortune smiled upon him at a math conference in Washington, DC, where he met Ken Ono, a prominent mathematician who had just joined Axiom, an artificial intelligence startup co-founded by his mentee Carina Hong.
After hearing Chen discuss the problematic theorem, Ono returned the next day with a proof derived from Axiom’s AI, AxiomProver. “Everything fell into place naturally after that,” remarked Chen, who subsequently collaborated with Axiom to publish the proof on arXiv, a repository for academic work.
Axiom’s AI had discovered a link between Chen and Gendron’s conjecture and a mathematical phenomenon from the 19th century, leading to a proof that it also verified itself. Ono noted, "What AxiomProver found was something that all the humans had missed."
In recent weeks, Axiom has claimed that its AI produced solutions to other previously unsolved mathematical questions. While it hasn’t tackled the most prominent problems in mathematics, it has resolved queries that have stumped experts for years, showcasing AI’s evolving mathematical capabilities. Several mathematicians have started applying AI tools to explore novel ideas and address existing problems.
Axiom’s advanced techniques might extend beyond pure mathematics, potentially aiding in developing resilient software against cybersecurity threats, verifying that code is reliably trustworthy.
“Math is really the great test ground and sandbox for reality,” stated Hong. She expressed that there are significant practical applications of their work with high commercial value.
Axiom’s strategy integrates large language models with AxiomProver, designed to logically work through mathematical problems to find provably correct solutions. In a related development, Google showcased a similar concept with its AlphaProof system in 2024. According to Hong, AxiomSolver features several notable advancements.
Ono emphasized the importance of the AI-generated proof for the Chen-Gendron conjecture in illustrating AI’s ability to assist mathematicians significantly. “This is a new paradigm for proving theorems,” he stated.
Unlike standard AI models, AxiomProver can independently verify proofs using a specialized mathematical language called Lean. This capability allows it to develop innovative methods for solving problems rather than merely sifting through existing literature.
Another new proof generated by AxiomProver independently tackled Fel’s Conjecture, relating to syzygies, mathematical expressions where numbers align in algebra. This conjecture references formulas from the notebook of the celebrated Indian mathematician Srinivasa Ramanujan, over a century old. Impressively, AxiomProver not only contributed to this theorem but crafted the entire proof independently.
Scott Kominers, a Harvard Business School professor aware of the conjecture and Axiom’s technology, expressed his astonishment: "It’s not just that AxiomProver managed to solve a problem like this fully automated, and instantly verified, which on its own is amazing, but also the elegance and beauty of the math it produced."
In addition to the Chen-Gendron proof, Axiom’s AI has also explored other complex mathematical models, including a probabilistic view of “dead ends” in number theory and concepts based on the tools created to solve the renowned Fermat’s Last Theorem.
Ono is hopeful that AxiomProver will not only aid mathematicians but also enhance our understanding of the discovery process itself. “I’m interested in trying to understand if you can make these aha moments predictable,” he noted.
Reflecting on his conjecture’s resolution, Chen conveyed optimism about AI’s role in mathematics. He said, “Mathematicians did not forget multiplication tables after the invention of the calculator. I believe AI will serve as a novel intelligent tool—or perhaps an ‘intelligent partner’ is more apt—opening up richer and broader horizons for mathematical research.”
