A new AI mathematics startup has solved 4 previously unsolved problems

five years ago, Mathematicians Dawei Chen and Quentin Gendron were trying to solve a difficult area of ​​algebraic geometry involving differentials, the elements of calculus used to measure distance along Curved surfaces. While working on a theory, they encountered an unexpected obstacle: their argument depended on a strange formula of Number theoryBut they could not solve or justify it. In the end, Chen and Gendron wrote a paper presenting their idea as a guess, not a theory.

Chen recently spent hours prodding ChatGPT in hopes of getting the AI ​​to come up with a solution to the yet-to-be-solved problem, but was unsuccessful. Then, during a reception at a mathematics conference in Washington, D.C., last month, Chen met Ken Ono, a well-known mathematician who had recently left his job at the University of Virginia to join Axiomthat artificial intelligence A startup co-founded by one of his students, Karina Hong.

Chen told Ono about the problem, and the next morning, Ono gave him a clue, thanks to the math-solving AI at his startup, AxiomProver. “Everything fell into place naturally after that,” says Chen, who worked with Axiom to write the guide, which is now ready. Published on arXiva public repository for academic papers.

Axiom’s AI tool found a link between the problem and a numerical phenomenon first studied in the 19th century. She then devised a clue, which she helpfully verified herself. “What AxiomProver found was something that all humans had missed,” Ono tells WIRED.

The clue is one of several solutions to unsolved mathematical problems that Axiom says its system has come up with in recent weeks. Artificial intelligence has not yet been able to solve any of the most famous (or lucrative) problems in mathematics, but it has found answers to questions that have puzzled experts in various fields for years. The evidence is evidence of the steady progress of AI capabilities in mathematics. In recent months, other mathematicians have reported using AI tools to explore new ideas and solve existing problems.

The techniques developed by Axiom may be useful outside the world of advanced mathematics. For example, the same techniques can be used to develop software that is more resilient to certain types of cybersecurity attacks. This may involve using artificial intelligence to verify that the code is reliable and trustworthy.

“Math is really a great testing ground and sandbox for reality,” says Hong, CEO of Axiom. “We think there are a lot of very important use cases with high business value.”

Axiom’s approach involves combining large language models with a proprietary AI system called AxiomProver that has been trained to reason through math problems to arrive at provable solutions. In 2024, Google presented a similar idea with A system called AlphaProof. Hong says AxiomSolver incorporates many important advances and newer technologies.

Ono says the AI-generated proof of Chen Gendron’s conjecture shows how AI can now usefully assist professional mathematicians. “This is a new model for proving theories,” he says.

The Axiom system is more than just an ordinary AI model, as it is able to verify proofs using a specialized mathematical language called Lean. Instead of simply searching the literature, this allows AxiomProver to develop new ways of solving problems.

A new proof created by AxiomProver shows how artificial intelligence is capable of solving mathematical problems entirely on its own. This evidence, which is also described in the paper Published on arXivprovides a solution to Feil’s conjecture, which concerns symmetries, or mathematical expressions where numbers line up in algebra. Remarkably, this conjecture includes formulas first found in the notebook of the legendary Indian mathematician Srinivasa Ramanujan More than 100 years ago. In this case, AxiomProver not only filled in the missing piece of the puzzle, but created the guide from beginning to end.