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The planar unit distance problem is about how many equal-sized lines you can draw that connect dots on an infinite sheet of paper<\/p>\n
Noga Alon et al. 2026, Open AI<\/p>\n<\/p>\n
An 80-year-old maths conjecture that has eluded the world\u2019s 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\u2019s mathematical ability.<\/p>\n
\u201cThis is a problem that I didn\u2019t expect to see solved in my lifetime,\u201d says Misha Rudnev<\/a> at the University of Bristol, UK. \u201cIt\u2019s absolutely a bomb.\u201d<\/p>\n Tim Gowers<\/a> at the University of Cambridge wrote that the solution is \u201ca milestone in AI mathematics\u201d in a blog post accompanying the work<\/a>. \u201cIf 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.\u201d<\/p>\n Twentieth-century mathematician Paul Erd\u0151s<\/a> considered the puzzle, known as the planar unit distance problem, as his \u201cmost striking contribution to geometry\u201d, 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?<\/p>\n Erd\u0151s conjectured that the patterns that yielded the most connections were points arranged in a grid, meaning the maximum number of connections would be only slightly higher than the number of points themselves. 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 Erd\u0151s\u2019s conjecture was more than 40 years ago.<\/p>\n Now, a model from OpenAI has found that Erd\u0151s was significantly wrong, and that you can arrange points in less symmetric patterns that can yield a far greater number of pairs.<\/p>\n \u201cMy immediate reaction was disbelief,\u201d says Will Sawin<\/a> at Princeton University. \u201cI thought the way that it was trying to solve it wouldn\u2019t 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.\u201d<\/p>\n OpenAI hasn\u2019t said exactly how the model differs from publicly available AIs or how it was trained, but OpenAI researcher Sheryl Hsu has publicly commented<\/a> that the model is \u201cgeneral purpose\u201d and wasn\u2019t trained \u201cwith the goal of doing math research\u201d.<\/p>\n 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.<\/p>\n \u201cThe 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,\u201d says Kevin Buzzard<\/a> at Imperial College London.<\/p>\n While the result is impressive, it is also partly a consequence of the fact that mathematicians didn\u2019t even consider that Erd\u0151s\u2019s original conjecture may have been false, says Samuel Mansfield<\/a> 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. \u201cThis is something that requires you to know a lot about multiple areas,\u201d he says. \u201cIn retrospect, it\u2019s maybe not so surprising. This seems to be what an AI would absolutely be good at doing.\u201d<\/p>\n The main appeal of the problem was the \u201cpure intellectual challenge\u201d, 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.<\/p>\n \u201cLike many other AI breakthroughs, it did not take humans long at all to internalise, understand and generalise the arguments,\u201d says Buzzard. \u201cOne can contrast this with some human breakthroughs which have taken the community months or years to validate.\u201d<\/p>\n Topics:<\/p>\n\n