An 80-year-old maths conjecture that has eluded the world鈥檚 greatest mathematicians has been cracked by an artificial intelligence model built by OpenAI. The result has stunned experts and is being hailed as a seismic moment for AI鈥檚 mathematical ability.
鈥淭his is a problem that I didn鈥檛 expect to see solved in my lifetime,鈥� says at the University of Bristol, UK. 鈥淚t鈥檚 absolutely a bomb.鈥�
at the University of Cambridge wrote that the solution is 鈥渁 milestone in AI mathematics鈥� in a . 鈥淚f 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.鈥�
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Twentieth-century mathematician Paul Erd艖s considered the puzzle, known as the planar unit distance problem, as his 鈥渕ost striking contribution to geometry鈥�, because it was seemingly simple to explain but deeply complex to answer. He asked: if you take an infinite-sized piece of paper and draw a number of dots in a pattern of your choice, what is the maximum number of equal-sized lines you can draw between these dots?
Erd艖s conjectured that the patterns that yielded the most connections were points arranged in a grid, meaning the maximum number of connections would grow only slightly larger than the number of points themselves, as you added more points to the grid. Successive attempts to prove that this really is the upper limit, or find a different arrangement of points that might lead to many more connections, yielded only small successes. The most recent improvement to The most recent improvement to Erd艖s鈥檚 conjecture was more than 40 years ago.
Now, a model from OpenAI has found that Erd艖s was significantly wrong, and that you can arrange points in less symmetric patterns that can yield a far greater number of pairs.
鈥淢y immediate reaction was disbelief,鈥� says at Princeton University. 鈥淚 thought the way that it was trying to solve it wouldn鈥檛 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.鈥�
OpenAI hasn鈥檛 said exactly how the model differs from publicly available AIs or how it was trained, but OpenAI researcher Sheryl Hsu has that the model is 鈥済eneral purpose鈥� and wasn鈥檛 trained 鈥渨ith the goal of doing math research鈥�.
The AI borrowed a technique from algebraic number theory to construct vast lattices in much higher dimensions than the two of a plane. Once it had identified and built these more complex shapes, it then collapsed them down to two dimensions, producing a shadow of the higher-dimensional shapes.
鈥淭he 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,鈥� says at Imperial College London.
While the result is impressive, it is also partly a consequence of the fact that mathematicians didn鈥檛 even consider that Erd艖s鈥檚 original conjecture may have been false, says at the University of Manchester, UK. Even if mathematicians did experiment with disproving it, very few geometry specialists would have then been knowledgeable enough in advanced number theory to do so. 鈥淭his is something that requires you to know a lot about multiple areas,鈥� he says. 鈥淚n retrospect, it鈥檚 maybe not so surprising. This seems to be what an AI would absolutely be good at doing.鈥�
The main appeal of the problem was the 鈥減ure intellectual challenge鈥�, says Rudnev, and it may not have any particular ramifications for other outstanding problems, but it has already sparked some further work. After seeing the proof, Sawin used the technique that the AI had discovered to produce a slightly improved, higher number for how many points could be joined together.
鈥淟ike many other AI breakthroughs, it did not take humans long at all to internalise, understand and generalise the arguments,鈥� says Buzzard. 鈥淥ne can contrast this with some human breakthroughs which have taken the community months or years to validate.鈥�
Article amended on 10 June 2026
We clarified Paul Erd艖s鈥檚 conjecture regarding the planar unit distance problem.
