OpenAI's reasoning model just cracked an 80-year-old math problem

How many pairs of points can be exactly distance 1 apart if you scatter n points on a flat plane?

That simple question, first posed by Paul Erdős in 1946, has puzzled mathematicians for nearly 80 years. Brass, Moser, and Pach called it possibly the best known and simplest to explain problem in discrete geometry, and Erdős even put a cash bounty on it. The long-standing consensus was that square-grid setups were basically optimal, meaning you could never achieve a true polynomial leap beyond near-linear growth.

OpenAI just upended that assumption. An internal general-purpose reasoning model autonomously disproved the conjecture, providing an infinite family of configurations that deliver a polynomial improvement.

Algebraic number theory crashes discrete geometry

For decades, Erdős's bound conjectured the maximum pairs at n^(1+o(1)), essentially matching scaled square grids using Gaussian integers.

The OpenAI model completely shattered that ceiling. It constructed configurations yielding at least n^(1+δ) unit-distance pairs for a positive exponent δ. While the AI proof left the exponent implicit, Princeton mathematics professor Will Sawin quickly refined the argument to nail down δ = 0.014.

What truly stunned experts was how the model got there. Instead of standard geometric tweaks, it reached deep into algebraic number theory, deploying sophisticated concepts like infinite class field towers and Golod–Shafarevich theory to establish the existence of number fields with the required symmetries.

A team of external mathematicians verified the proof and published a companion paper explaining the result. Fields medalist Tim Gowers called it a milestone in AI mathematics, while leading number theorist Arul Shankar noted the model showed it is capable of original ingenious ideas rather than acting as a mere assistant.

Crucially, this was not a narrow system hardcoded for geometry proofs or equipped with specialized symbolic search scaffolds. OpenAI prompted its standard general-purpose reasoning model on a batch of Erdős open problems, and it found the proof on its own.

Why it matters

Mathematics is the ultimate stress test for reasoning because hallucinations are fatal. A 50-step argument with one bogus assertion is worthless. By maintaining logical coherence across a deep proof and bridging two wildly distant mathematical fields, AI just passed a threshold many thought was years away.

For researchers, this flips the narrative around AI in science. The model did not just crunch data or autocomplete Lean code; it introduced unexpected conceptual machinery that human experts had overlooked for eight decades. As mathematician Thomas Bloom pointed out, it opens an entirely new playbook for discrete geometry.

We think the real takeaway goes far beyond geometry. The ability to autonomously synthesize deep ideas across siloed domains is exactly what is needed in drug discovery, materials science, and fundamental physics. While human experts are still needed to interpret and refine the output, the era of AI as a genuine creative co-discoverer has officially arrived.