An OpenAI model has disproved a central conjecture in discrete geometry that has remained unsolved for nearly 80 years. The conjecture, known as the planar unit distance problem, asked how many pairs of points can be exactly one unit apart when placing n points in the plane. First posed by Paul Erdős in 1946, the problem has been a focus of mathematical research, with the prevailing belief that square grid constructions were essentially optimal for maximizing unit-distance pairs. OpenAI’s internal model, however, provided an infinite family of examples that achieve a polynomial improvement over these constructions. The proof, which has been verified by external mathematicians, introduces sophisticated ideas from algebraic number theory to an elementary geometric question. *Source: [openai](https://openai.com/index/model-disproves-discrete-geometry-conjecture/)* The result is notable for how it was discovered. Unlike systems trained specifically for mathematics or tailored to the unit distance problem, the proof came from a general-purpose reasoning model. As part of a broader effort to test AI’s potential in frontier research, the model was evaluated on a collection of Erdős problems. In this case, it produced a proof resolving the open problem. The method used to solve the problem is also significant, as it applies unexpected and advanced concepts from algebraic number theory to a seemingly simple geometric question. This marks the first time that a prominent open problem in mathematics has been solved autonomously by AI. The proof is available for review, along with a companion paper by leading external mathematicians explaining the argument and its significance. *Source: [openai](https://openai.com/index/model-disproves-discrete-geometry-conjecture/)* Previously, the best known construction for maximizing unit-distance pairs came from a rescaled square grid, which achieved a growth rate of n^{1 + C / log log(n)} for a constant C. This rate was considered essentially optimal for decades. However, the new result shows that for infinitely many values of n, configurations of n points can achieve at least n^{1+δ} unit-distance pairs, where δ is a fixed positive exponent. While the original AI proof does not specify δ, a forthcoming refinement by Princeton professor Will Sawin has demonstrated that δ can be as small as 0.014. This breakthrough challenges the long-standing belief that square grid constructions were the best possible and opens new avenues for research in discrete geometry. *Source: [openai](https://openai.com/index/model-disproves-discrete-geometry-conjecture/)*