OpenAI AI Model Cracks 80-Year-Old Math Problem, But Proof Still Incomplete

OpenAI has announced a new advance in artificial intelligence reasoning, with its technology solving a long-standing mathematical problem first posed 80 years ago.

The company behind ChatGPT said it achieved a breakthrough on the planar unit distance problem, a challenge introduced by Hungarian mathematician Paul Erdős in 1946.

Erdős’s question is straightforward: Given a set of points on a flat surface, how many pairs can be exactly the same distance apart? He conjectured that the number would grow only slightly faster than the number of points themselves.

OpenAI’s model reached a different conclusion, drawing on multiple branches of mathematics to identify a family of point arrangements that exceed the limit Erdős proposed.

“For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids,” OpenAI wrote on X. “An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better.”

While the result has generated excitement among mathematicians, the broader problem remains unsolved. The AI did not provide a new estimate for how quickly the number of distance pairs increases; it only demonstrated that Erdős’s original limit was too low.

OpenAI, which is preparing for a U.S. stock market listing, said the calculations were performed by a general-purpose reasoning model that breaks problems into smaller steps, rather than a system specifically trained for mathematics.

The startup has previously stumbled in attempts to solve Erdős’s problems, including a purported breakthrough last year that turned out to be based on existing literature the model had absorbed. This time, OpenAI’s work has been validated by mathematicians, including Thomas Bloom, who maintains the Erdős problems website and had criticized the earlier claim.

Bloom co-authored a companion paper to OpenAI’s blog post highlighting the achievement. He wrote that the AI system obtained its results by “persevering down paths that a human may have dismissed as not worth their time to explore.”

However, Bloom noted that humans played a crucial role in the process. “While the original proof produced by AI was completely valid, it was significantly improved by the human researchers at OpenAI and the many other mathematicians involved in the present paper,” he wrote.

Mathematician Tim Gowers, also contributing to the companion paper, described the result as “a milestone in AI mathematics.”

Andrew Rogoyski, of the Institute for People-Centred AI at the University of Surrey, said the announcement demonstrates that AI is providing humans with new ways to approach problems. “It’s becoming clear that AI is impacting the world of creative thought and will become a fundamental tool of future scientific research,” he said.

The advance marks a step forward in AI’s ability to contribute to pure mathematics, though human mathematicians remain essential for refining and verifying the results.

OpenAI’s work suggests that AI models can explore unconventional approaches that may elude human intuition, but the final resolution of the Erdős problem still awaits further breakthroughs.