An OpenAI model solved a famous math problem that stumped humans for 80 years

In May, OpenAI announced that an AI model had disproved the Erdős unit distance conjecture, a problem in discrete geometry that had stumped mathematicians for 80 years. Tim Gowers, a Fields Medal winner, called it a "milestone in AI mathematics." Daniel Litt, a University of Toronto professor, noted it as the "first example of a result produced autonomously by an AI that I find exciting." This is arguably the first time an AI system has resolved a major open conjecture. While impressive, it represents a continuation of AI progress in mathematics rather than a radical departure. Three years ago, LLMs struggled with arithmetic; last year, they excelled in high school math competitions. Early this year, AI began contributing to mathematical research in constrained settings, requiring significant human interpretation. OpenAI's new result advances this progression. The AI model proficiently applied existing ideas from various mathematical subfields to construct a comprehensive proof. However, it did not introduce any entirely new techniques. Human mathematicians have since refined and expanded upon the initial findings.

This development suggests a future where human mathematicians and AI models collaborate. AI possesses a broader knowledge of past work and a greater capacity to pursue tedious proof strategies that might not yield results. Conversely, humans retain the ability to delve deeper into specific problems and formulate more insightful questions. However, the rapid advancement of AI in mathematics raises questions about the future role of human mathematicians within the next decade.

Paul Erdős, one of history's most prolific mathematicians, published over 1,500 papers. He was renowned for posing problems that were simple to state but had profound implications. In 1946, he introduced the unit distance problem, which involves arranging points in a 2D plane and determining the maximum number of pairs exactly one unit apart. As the number of points increases, finding an exact solution becomes exceedingly complex.

Consequently, Erdős focused on establishing upper and lower bounds for the number of unit distances for a given number of points. To calculate a lower bound, he hypothesized an arrangement of points in a grid. He observed that by adjusting the grid spacing, one could increase the number of unit distances by incorporating diagonals. For instance, if the grid spacing was 1/√65, each point would be one unit away from 16 other points due to the Pythagorean theorem, which relates the sides of a right triangle (a² + b² = c²). By selecting c² carefully, more whole-number diagonals can be incorporated, leading to a greater number of unit-distance pairs.