An OpenAI model disproved an 80-year-old Erdős geometry conjecture; mathematicians checked it
An unreleased OpenAI model found a counterexample few experts expected. Mathematicians checked it the same day; one called it AI's best yet.
In 1946 Paul Erdős made a guess about points on a flat surface. Among n points, he conjectured, the number of pairs exactly one unit apart can grow only slightly faster than n. In effect, grid-like arrangements would be about the best possible. Most mathematicians believed it.
On 20 May 2026 OpenAI announced that an internal, unreleased general-purpose reasoning model had found arrangements that beat that limit, using tools from algebraic number theory. The same day, nine mathematicians, including Noga Alon, Thomas Bloom, Tim Gowers and Melanie Matchett Wood, posted a short, human-verified version on arXiv. Will Sawin made the bound explicit: more than n^1.014 unit distances.
Gowers said no previous AI-generated proof “has come close to that.” Bloom called it the most impressive achievement of AI in mathematics so far. The proof was later formalised in Lean, so a computer has checked it. Mathematicians then used related techniques to disprove a real-number version of another conjecture, the sum-product conjecture.
What it means
An AI settled a well-known question that experts had believed for 80 years, and leading mathematicians confirmed the result. The improvement itself is small, and the exact answer to the unit distance problem is still unknown. Researchers have already reused its ideas on another problem.
Caveats
- The model was unreleased; compute and any human involvement were not disclosed in the sources we could read.
- The verifying mathematicians note that, in hindsight, the key ideas can be traced to earlier work in number theory.
- The improvement is a small exponent; the known upper bound is unchanged.