OpenAI Model Autonomously Disproves 80-Year-Old Erdős Geometry Conjecture

For nearly 80 years, mathematicians have studied the planar unit distance problem first posed by Paul Erdős in 1946: given n points in the plane, what is the maximum number of pairs at exactly unit distance apart? The prevailing belief was that rescaled square grid constructions were essentially optimal. An internal OpenAI model has now disproved this conjecture, providing an infinite family of examples that yield a genuine polynomial improvement over the grid-based approach.

What makes this result extraordinary is not just the mathematics — which brings sophisticated ideas from algebraic number theory to an elementary geometric question — but how it was found. The proof was produced autonomously by a general-purpose reasoning model, not a specialized math system. It was tested on a collection of Erdős problems and independently generated the solution.

External mathematicians verified the proof and wrote a companion paper. Fields Medalist Tim Gowers called it 'a milestone in AI mathematics,' while number theorist Arul Shankar stated that 'current AI models go beyond just helpers to human mathematicians — they are capable of having original ingenious ideas, and then carrying them out to fruition.' This marks the first time a prominent open problem central to a subfield of mathematics has been solved autonomously by AI, signaling a genuine shift in the role of AI in frontier research.