Paul Erdős, who published more than 1,500 papers during his lifetime, also left a legacy of more than 1,000 open research questions, some of which are now being solved with AI. Credit: George Csicsery
An 80-year-old challenge in geometry has been cracked by mathematicians working for the tech firm Open AI using a single prompt from an AI chatbot.
The company has not revealed all the precise details and steps of how it did this, nor the name of the AI system that achieved the result, which it has published on its website. However, the finding has been verified independently by mathematicians not connected to the firm.
OpenAI announced on 20 May that its chatbot software had disproved Paul Erdős (1913–1996) on what is called the unit-distance problem. In 1946, Erdős worked out what he suggested was the best arrangement of points on a plane so that as many pairs as possible are at a given distance from each other – and he put down a challenge: no one could do better.
Now, OpenAI says that its system has done precisely that. It did so by using techniques in algebraic number theory, which enabled it to choose points with coordinates that were the solutions of particular equations. And the finding has astonished mathematicians.
“If Erdős were alive, I am sure that he would just be raving about this advance,” says Tom Trotter, a mathematician at the Georgia Institute of Technology in Atlanta who co-authored papers with the late Erdős.
Sebastien Bubeck, a mathematician at OpenAI in San Francisco, California, says he believes this is the first time that AI has autonomously produced an important result in any field of research. And Tony Feng, a mathematician at the University of California at Berkeley, wrote on X: “I like to think that I have been a relatively measured voice on the impact of AI on mathematics, but this is incredible.”
Daniel Litt, a mathematician at the University of Toronto in Canada and one of the independent researchers OpenAI called upon to verify the proof, says that this “the first result produced autonomously by an AI that I find interesting in itself”.
Age-old problem
In geometry, points can be arranged on a plane so that many pairs have the same mutual distance. For example, a regular polygon with nine edges has nine such pairs of points, because all nine edges have the same length. Putting nine points on a square grid gives 12 such pairs. Erdős showed how larger and larger grids could contain a number of same-distance points that grew to infinity ever-so-slightly faster than did the number of points. Moreover, he conjectured that no one could find a better method to arrange such a large number of same-distance points.
Credit: Alvaro Lozano-Robledo
Except that OpenAI says it now has. The company’s AI model did so using techniques in algebraic number theory, which enabled it to choose points with coordinates that were the solutions of particular equations. “The model has a produced a single, very long chain of thought,” says Bubeck. He adds that the prompt that led to this answer was an open-ended question about whether Erdős’s conjecture could be true or false, not an explicit request to prove him wrong. “It’s kind of remarkable to see the model really reasoning through the problem like a human,” says OpenAI mathematician Mehtaab Swahney.
Closed AI
This reasoning is contained in a 125-page document, which the company has not fully released. It is also not disclosing the name of its model. Bubeck says that it is an experimental, general-purpose reasoning model — not one specifically designed to solve maths problems — and that it did all the work autonomously, in response to a single prompt and was a machine-rewritten statement of Erdős’s question.
