概要
公開日: 2026-05-20
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原文
May 20, 2026 Research Milestone An OpenAI model has disproved a central conjecture in discrete geometry Read the proof (opens in a new window) Read the companion remarks (opens in a new window) Loading… Share For nearly 80 years, mathematicians have studied a deceptively simple question: if you place n n n points in the plane, how many pairs of points can be exactly distance 1 1 1 apart? This is the planar unit distance problem, first posed by Paul Erdős in 1946. It is one of the best-known questions in combinatorial geometry, easy to state and remarkably difficult to resolve. The 2005 book Research Problems in Discrete Geometry , by Brass, Moser, and Pach, calls it “possibly the best known (and simplest to explain) problem in combinatorial geometry.” Noga Alon, a leading combinatorialist at Princeton, describes it as “one of Erdős’ favorite problems.” Erdős even offered a monetary prize for resolving this problem. Today, we share a breakthrough on the unit distance problem. Since Erdős’s original work, the prevailing belief has been that the “square grid” constructions depicted further below were essentially optimal for maximizing the number of unit-distance pairs. An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement. The proof has been checked by a group of external mathematicians. They have also written a companion paper explaining the argument and providing further background and context for the significance of the result. The result is also notable for how it was found. The proof came from a new general-purpose reasoning model, rather than from a system trained specifically for mathematics, scaffolded to search through proof strategies, or targeted at the unit distance problem in particular. As part of a broader effort to test whether advanced models can contribute to frontier research, we evaluated it on a collection of Erdős problems. In this case, it produced a proof resolving the open problem. This proof is an important milestone for the math and AI communities. It marks the first time that a prominent open problem, central to a subfield of mathematics, has been solved autonomously by AI. It also demonstrates the depth of reasoning these systems now support. Mathematics provides a particularly clear testbed for reasoning: the problems are precise, potential proofs can be checked, and a long argument only works if the reasoning holds together from beginning to end. The method by which the problem was solved is also notable. The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. Fields medalist Tim Gowers, writing in the companion paper, calls the result “a milestone in AI mathematics.” According to leading number theorist Arul Shankar, “In my opinion this paper demonstrates that current AI models go beyond just helpers to human mathematicians – they are capable of having original ingenious ideas, and then carrying them out to fruition”. Mathematicians on the result 1 of 4 “ This has been one of Erdős' favorite problems, I have heard him myself mentioning the problem multiple times in his lectures. I believe it would be fair to say that every mathematician working in Combinatorial Geometry thought about this problem, and lots of mathematicians working in other areas spent at least some time thinking about it… The solution of the problem by the internal model of Open AI is, in my opinion, an outstanding achievement, settling a long-standing open problem. The fact that the correct answer is not n 1 + o ( 1 ) n^{1+o(1)} n 1 + o ( 1 ) is surprising, and the construction and its analysis apply fairly sophisticated tools from algebraic number theory in an elegant and clever way. ” “ There is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics: if a human had written the paper and submitted it to the Annals of Mathematics and I