Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946.

For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids.

An OpenAI model has now disproved that… pic.twitter.com/j2g3Ze0zEG

— OpenAI (@OpenAI) May 20, 2026

In 1946, Erdős proposed the following hypothesis: if n points are placed on a plane, how many pairs of points can be exactly at a distance of at least n1-δ(1).

It is considered one of the most famous problems in combinatorial geometry: it is simply formulated but has resisted solution for decades.

OpenAI stated that its internal model disproved this longstanding hypothesis in discrete geometry. It published separate material describing the result, along with links to proofs and accompanying notes.

The model found an infinite family of examples that provide a polynomial improvement over constructions previously considered close to optimal.

The work demonstrates the existence of a constant δ > 0 and infinitely many values of n, for which it is possible to construct configurations of n points with at least n1+δ pairs at distance 1.

Previously, the best known construction, based on scaled square grids, yielded approximately n(1 + C / log(log(n))) unit distances. This grows only slightly faster than linear: since log(log(n)) increases with n, the additional factor C / log(log(n)) gradually approaches zero.

Moreover, the solution did not come from geometry itself but from algebraic number theory. Instead of classical Gaussian integers of the form z = a + bi, where a and b are integers (including zero), and i is the imaginary unit, the model used more complex number fields with rich symmetries.

The proof employs tools such as infinite towers of class fields and the Golod–Shafarevich theorem. For number theory specialists, these are well-known methods, but their connection to an elementary geometric problem proved to be unexpected.

Independent Audit

OpenAI stated that the proof was verified by a group of external mathematicians. The company also emphasized that the result was obtained not by a specialized mathematical system but by a general-purpose reasoning model.

According to the startup, the work was part of a broader effort to test whether advanced neural networks can contribute to cutting-edge scientific research.

The OpenAI material includes assessments from several mathematicians. In particular, Fields Medalist Timothy Gowers called the result “a milestone for AI in mathematics.” Also cited are words from Arul Shankar, a mathematician at the University of Toronto, who stated that current models are capable not only of assisting but also of proposing original ideas and carrying them through to results.

Recall that in February, Google DeepMind’s division introduced the AI agent Aletheia, which set a new record in the IMO-ProofBench Advanced benchmark.

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