Washington, United States. OpenAI has revealed that one of its internal artificial intelligence models found a counterexample to a famous conjecture made by Hungarian mathematician Paul Erdős in 1946. The result has drawn attention from the mathematical community and led to further advances by other researchers.
Planar unit distance problem
The problem, known as the planar unit distance problem or Erdős problem 90, asks how many pairs of points can be placed exactly one unit apart among a given number of points on an infinite plane.
Canadian mathematician Daniel Litt described the finding as “the first result produced autonomously by an AI that I find interesting in itself”.
AI and further results
The breakthrough came from a general-purpose AI model rather than one designed specifically for mathematics. Days after OpenAI’s paper, US mathematician Will Sawin used the same line of reasoning to obtain an improved result.
Last week, a team from Google DeepMind also reported that one of its models resolved nine lesser open problems left by Erdős.
Longstanding conjecture
Erdős was one of the most prolific mathematicians of the twentieth century and was known for posing simple questions that resisted solutions for decades.
The unit distance problem appears straightforward at first. It asks how to arrange n points so that the number of pairs exactly one unit apart is maximized.
Earlier thinking
A square grid is one natural arrangement that seems promising, since its spacing creates many pairs at regular distances. This idea shaped much of the early thinking on the problem.
As the number of points increases, grid-like arrangements have continued to appear highly effective.
Research history
For decades, mathematicians believed such regular structures were close to optimal. Erdős conjectured that no construction could substantially improve on them, even for very large numbers of points.
The new best result by Sawin reportedly begins to show improvements only at around 102000000 points, described as a one followed by two million zeroes.
Over the past 80 years, attempts to prove or disprove the conjecture have connected the problem with incidence geometry, graph theory and extremal combinatorics. A full proof has remained out of reach, although many mathematicians had come to believe the conjecture was probably true.
