OpenAI’s AI Model Challenges Erdős’s 80-Year-Old Unit Distance Hypothesis

OpenAI's AI model challenges Erdős's 80-year-old hypothesis on unit distances.

OpenAI has announced a breakthrough in the classical mathematical problem posed by Paul Erdős regarding unit distances.

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 n^1-δ(1).

It is considered one of the most famous problems in combinatorial geometry: simple to state, yet unsolved for decades.

OpenAI claimed that its internal model has refuted the long-standing hypothesis in discrete geometry. The company published a separate document detailing the result, including links to proofs and accompanying notes.

The model discovered an infinite family of examples that provide a polynomial improvement over constructions previously thought to be near-optimal.

The work demonstrates the existence of a constant δ > 0 and infinitely many values of n, for which configurations of n points can be constructed with at least n^1+δ pairs at a distance of 1.

The best-known previous construction, based on a scaled square grid, provided approximately n^{(1 + C / log(log(n)))} unit distances. This is only slightly faster than linear growth: as log(log(n)) increases with n, the additional factor C / log(log(n)) gradually approaches zero.

Interestingly, the solution emerged not 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 utilized more complex number fields with rich symmetries.

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

Independent Audit

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

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

The OpenAI document includes assessments from several mathematicians. Notably, Fields Medalist Timothy Gowers described the result as a “milestone for AI in mathematics.” The document also quotes University of Toronto mathematician Arul Shankar, who stated that current models are capable not only of assisting but also of proposing original ideas and bringing them to fruition.

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