Mathematicians stunned by AI’s biggest breakthrough in mathematics yet

A longstanding mathematical conjecture, unsolved for 80 years, has been resolved by an AI model developed by OpenAI. This breakthrough has astonished experts and is seen as a pivotal moment in the mathematical capabilities of artificial intelligence.

“This is a problem that I didn’t expect to see solved in my lifetime,” remarks Misha Rudnev from the University of Bristol, UK. “It’s absolutely a bomb.”

Tim Gowers of the University of Cambridge described the solution as “a milestone in AI mathematics” in a blog post accompanying the work. “If a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that.”

Paul Erdős, a mathematician from the 20th century, regarded the planar unit distance problem as his “most striking contribution to geometry,” due to its deceptively simple premise yet complex solution. He posed the question: if you draw dots on an infinite sheet of paper, how many equal-length lines can connect them?

Erdős theorized that the highest number of connections would come from a grid arrangement of points, suggesting the maximum connections would be just above the number of points. Attempts to prove this upper limit or find a more connecting arrangement had minimal success, with the last significant progress occurring over 40 years ago.

OpenAI’s model has now revealed that Erdős’s theory was wrong, showing that less symmetric patterns can produce more pairs.

“My immediate reaction was disbelief,” says Will Sawin at Princeton University. “I thought the way that it was trying to solve it wouldn’t work, but then I looked at it more and I convinced myself that it does work. I pretty quickly became convinced this is the most significant achievement by AI in mathematics so far.”

OpenAI has not disclosed the specifics of how the model deviates from publicly available AIs or its training process. However, researchers have stated that the model is “general purpose” and not specifically trained for math research.

The AI utilized a technique from algebraic number theory to create large lattices in higher dimensions and then projected these into two dimensions, forming a complex shadow of the higher-dimensional shapes.

“The counterexample discovered by the AI is complex, and although the ideas to produce it were already in the literature, it certainly takes some ingenuity to put them together,” says Kevin Buzzard at Imperial College London.

While the outcome is impressive, it is partly because mathematicians hadn’t considered that Erdős’s original conjecture might be false, notes Samuel Mansfield at the University of Manchester, UK. Even if some tried to disprove it, not many geometry experts had the advanced number theory knowledge needed. “This is something that requires you to know a lot about multiple areas,” he says. “In retrospect, it’s maybe not so surprising. This seems to be what an AI would absolutely be good at doing.”

The problem’s main attraction was its “pure intellectual challenge,” according to Rudnev. It may not affect other major problems, but it has already inspired further research. After reviewing the proof, Sawin applied the AI’s technique to achieve a slightly better estimate of how many points could be interconnected.

“Like many other AI breakthroughs, it did not take humans long at all to internalise, understand and generalise the arguments,” says Buzzard. “One can contrast this with some human breakthroughs which have taken the community months or years to validate.”