An OpenAI model has disproved a central conjecture in discrete geometry

An OpenAI model has disproved a central conjecture in discrete geometry | OpenAI

May 20, 2026

An OpenAI model has disproved a central conjecture in discrete geometry

Read the proof Read the companion remarks

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For nearly 80 years, mathematicians have studied a deceptively simple question: if you place nn points in the plane, how many pairs of points can be exactly distance 11 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

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> “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 n1+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.”

Noga Alon

• Noga Alon
• Tim Gowers
• Arul Shankar
• Jacob Tsimerman

The proof is available here⁠. The companion paper by leading external mathematicians is available here⁠. You can find an abridged version of the model’s chain of thought here⁠.

Previously known construction of many unit distances from a rescaled square grid.

The unit distance problem

Let u(n)u(n) be the largest possible number of unit-distance pairs among nn points in the plane. Examples attaining linear growth rate are easy to construct: placing nn points in a line gives n−1n-1 pairs, while a square grid gives about 2n2n pairs. The previously best known construction, coming from a rescaled square grid, turns out to give even more: n1+C/log⁡log⁡(n)n^{1 + C / \log \log(n)} for a constant CC. Since log⁡log⁡(n)\log \log(n) tends to infinity with nn, the additional term in the exponent tends to 00, meaning these constructions achieve growth only slightly faster than linear. For decades, it was widely believed that this rate was essentially the best possible, and no construction could improve significantly over the square grid. In technical terms, Erdős conjectured an upper bound of n1+o(1)n^{1+o(1)} in which the additional o(1)o(1) indicates a term tending to 00 with nn.Our new result disproves this conjecture. More precisely, for infinitely many values of nn, the proof constructs configurations of nn points with at least n1+δn^{1+\delta} unit-distance pairs, for some fixed exponent δ>0\delta > 0. (The original AI proof does not give an explicit δ\delta, but a forthcoming refinement due to Princeton mathematics professor Will Sawin has shown one can take δ=0.014\delta=0.014.)The history of the problem helps to see why the result is surprising. The best known lower bound had been essentially unchanged since Erdős’s original 1946 construction. The best upper bound, O(n4/3)O(n^{4/3}), dates to work by Spencer, Szemerédi, and Trotter in 1984, and despite later refinements and related structural work by Székely, Katz and Silier, Pach, Raz, and Solymosi and by others, the upper bound has remained essentially unchanged. As evidence in favor of the conjecture, Matoušek and Alon-Bucić-Sauermann studied the problem with non-Euclidean distances in the plane, and proved that "most" of these non-Euclidean distances obey the conjecture in some sense.Surprisingly, the key ingredients of the construction come from a very different part of mathematics known as algebraic number theory, which studies concepts like factorization in extensions of the integers known as algebraic number fields.

After verifying the initial proof, we investigated the success rate of our models on this…