OpenAI Model Autonomously Solves 80-Year-Old Geometry Problem
OpenAI has announced that an internal general-purpose reasoning model has disproved a longstanding conjecture in discrete geometry — the Erdős unit distance problem — marking a historic milestone in AI’s ability to contribute to frontier mathematics.
First posed by Paul Erdős in 1946, the problem asks a deceptively simple question: given n points in the plane, what’s the maximum number of pairs that can be exactly distance 1 apart? The best known construction, a rescaled square grid, achieved a growth rate of n^{1 + C/log log n} — meaning the exponent term tends to zero as n grows. For decades, mathematicians widely believed this was essentially optimal.
The OpenAI model constructed an infinite family of configurations that achieve n^{1+δ} unit-distance pairs (Princeton’s Will Sawin subsequently refined δ = 0.014), disproving Erdős’s conjecture. Remarkably, the proof draws on deep tools from algebraic number theory — infinite class field towers and Golod-Shafarevich theory — establishing an entirely unexpected connection between abstract number theory and Euclidean geometry.
Fields Medalist Tim Gowers called the result “a milestone in AI mathematics.” Number theorist Arul Shankar noted: “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.”
This matters far beyond a single theorem. If a model can maintain coherence across a long mathematical argument, connect ideas across distant fields, and produce work that withstands expert scrutiny, those are capabilities transferable to physics, biology, materials science, and medicine — bringing us closer to automated research assistants that can genuinely accelerate scientific discovery.