Five years ago mathematicians Dawei Chen and Quentin Gendron tried to unravel a hard area of algebraic geometry involving differentials, elements of calculus used to measure distances along curved surfaces. While working on one theorem, they encountered an unexpected obstacle: their argument was based on a strange formula from number theory, but they were unable to solve or justify it. Ultimately, Chen and Gendron wrote a paper presenting their idea as a guess, not a claim.
Chen recently spent many hours prompting ChatGPT in hopes that the AI would find a solution to the still unresolved problem, but it didn’t work. Then last month, at a reception at a mathematics conference in Washington, Chen met Ken Ono, a renowned mathematician who had recently left his job at the University of Virginia to join Axioman artificial intelligence start-up co-founded by one of his protégés, Carina Hong.
Chen told Ono about the problem, and the next morning Ono presented him with proof, courtesy of his startup’s artificial intelligence, AxiomProver. “Everything fell into place naturally after that,” says Chen, who worked with Axiom to write the proof that is now published submitted to arXivpublic repository of scientific articles.
Axiom’s AI tool found a connection between the problem and a numerical phenomenon first studied in the 19th century. He then developed a proof that he verified himself. “What AxiomProver discovered everyone missed,” Ono tells WIRED.
The proof is one of several solutions to unsolved math problems that Axiom has come up with in recent weeks. Artificial intelligence has not yet solved any of the most notable (or lucrative) problems in mathematics, but it has found answers to questions that have puzzled experts in various fields for years. The evidence demonstrates the ever-evolving mathematical abilities of artificial intelligence. In recent months, other mathematicians have reported using artificial intelligence tools to discover up-to-date ideas and solve existing problems.
The techniques developed by Axiom may prove useful beyond the world of advanced mathematics. For example, the same approach can be used to develop software that is more resistant to certain types of cybersecurity attacks. This would involve using artificial intelligence to check whether the code is reliable and trustworthy.
“Mathematics is truly a great testing ground and sandbox of reality,” says Hong, CEO of Axiom. “We believe there are a number of very important use cases with high commercial value.”
Axiom’s approach involves combining huge language models with a proprietary artificial intelligence system called AxiomProver, which is trained to solve mathematical problems to find solutions that are provably correct. In 2024, Google demonstrated a similar idea with a system called AlphaProof. Hong says AxiomSolver includes several notable advances and newer techniques.
Ono says the AI-generated proof of the Chen-Gendron conjecture shows how AI can now significantly aid professional mathematicians. “It’s a new paradigm for proving theorems,” he says.
The Axiom system is more than just an AI model, as it is able to verify evidence using a specialized mathematical language called Lean. Instead of simply searching the literature, this allows AxiomProver to develop truly creative ways to solve problems.
Another of the up-to-date pieces of evidence generated by AxiomProver shows how artificial intelligence is able to solve math problems completely on its own. This proof, which was also described in the article submitted to arXivprovides a solution to Fel’s hypothesis, which concerns syzygy, i.e. mathematical expressions in which numbers in algebra are arranged as one whole. Interestingly, this assumption concerns patterns that were first found in the notebook of the legendary Indian mathematician Srinivasa Ramanujan over 100 years ago. In this case, AxiomProver not only filled in the missing piece of the puzzle, but devised the proof from start to finish.