A Fresh AI Mathematics Startup Has Successfully Solved Four Previously Intractable Problems.

Five years ago, mathematicians Dawei Chen and Quentin Gendron were delving into a challenging segment of algebraic geometry that involved differentials, which are calculus elements used for measuring distances across curved surfaces. While tackling a particular theorem, they encountered an unexpected hurdle: their reasoning relied on a peculiar formula from number theory, which they couldn’t solve or validate. Ultimately, Chen and Gendron published a paper framing their insight as a conjecture instead of a theorem.

Recently, Chen dedicated hours to engaging ChatGPT, seeking assistance with the unresolved issue, but to no avail. However, during a reception at a mathematics conference in Washington, DC, last month, he met Ken Ono, a prominent mathematician who had recently transitioned from the University of Virginia to Axiom, an artificial intelligence startup co-founded by his former mentee, Carina Hong.

Chen shared the problem with Ono, and the next day, Ono provided him with a proof, thanks to Axiom’s math-solving AI, AxiomProver. “Everything fell into place naturally after that,” remarks Chen, who collaborated with Axiom to document the proof, which has now been made available on arXiv, a public repository for academic papers.

Axiom’s AI tool identified a link between the problem and a numerical phenomenon first explored in the 19th century. It subsequently formulated a proof, which it verified on its own. “What AxiomProver discovered was something that all the humans had overlooked,” Ono tells WIRED.

This proof represents one of several resolutions to unresolved mathematical problems that Axiom claims its system has generated recently. While the AI hasn’t tackled any of the most renowned (or lucrative) mathematical challenges, it has resolved questions that have perplexed experts for years in various fields. These proofs illustrate AI’s continuously growing capabilities in mathematics. Recently, other mathematicians have also reported utilizing AI tools to investigate new concepts and resolve existing issues.

The methods being developed by Axiom may have applications beyond advanced mathematics. For instance, similar techniques could be employed to create software that offers increased resistance to specific types of cybersecurity threats. This would require using AI to ensure that the code is demonstrably reliable and trustworthy.

“Math is truly the ultimate testing ground and sandbox for reality,” states Hong, Axiom’s CEO. “We believe there are numerous significant use cases with high commercial potential.”

Axiom’s strategy combines large language models with a proprietary AI system called AxiomProver, which is designed to reason through mathematical problems to produce solutions that are demonstrably accurate. In 2024, Google showcased a similar concept with a system named AlphaProof. Hong asserts that AxiomSolver integrates several major advancements and newer methodologies.

Ono notes that the AI-generated proof for the Chen-Gendron conjecture exemplifies how AI can now effectively assist professional mathematicians. “This is a new paradigm for theorem proving,” he explains.

Axiom’s system goes beyond a standard AI model, as it can verify proofs using a specialized mathematical language known as Lean. This approach enables AxiomProver to create genuinely novel solutions rather than merely searching through existing literature.

Another recent proof generated by AxiomProver illustrates the AI’s capability to solve mathematical problems independently. This proof, also detailed in a paper on arXiv, provides a solution to Fel’s Conjecture, which involves syzygies—mathematical expressions where numbers align in algebra. Notably, this conjecture incorporates formulas first documented in the notebook of the legendary Indian mathematician Srinivasa Ramanujan over a century ago. In this instance, AxiomProver didn’t simply fill a gap in the reasoning; it constructed the proof from the ground up.