A general-purpose reasoning model from OpenAI has overturned a conjecture about unit distances on a plane that Paul Erdős posed back in 1946. The original conjecture claimed that if you place n points in the plane, the number of pairs that are exactly 1 unit apart can't significantly exceed n^{1+o(1)}. Mathematicians long thought that grid-like constructions were close to the limit.
What makes this proof remarkable is that the model wasn't a specialized math engine, nor did it rely on search scaffolding built specifically for this problem. It came up with a new family of point configurations that push the number of unit-distance pairs up to n^{1+δ} (with δ a positive constant), shattering the upper bound that had held for nearly 80 years.
The solution crosses disciplinary boundaries. The model brings deep tools from algebraic number theory into the combinatorial geometry of the Euclidean plane. After checking the raw proof, external mathematicians — including Noga Alon, Tim Gowers, and Arul Shankar — posted an accompanying paper on arXiv explaining that the derivation relies on sophisticated algebraic number theory ideas like those of Ellenberg-Venkatesh and Golod-Shafarevich.
Fields Medalist Tim Gowers called the result a milestone for AI in mathematics. Number theorist Arul Shankar believes the model has already shown it can generate original ideas and carry through a complete argument. More than the conclusion itself, the bigger technical advance is that a general-purpose model, without a custom toolchain, independently traced a path that human experts had missed for decades.