Introduction to the Problem
An OpenAI model has solved an 80-year-old math problem that had long been out of reach for the world's greatest scientists. The result of the work by a team led by Noga Alon has shaken the scientific community.
The Planar Unit Distance Problem
The problem is stated as follows: if you take an infinite sheet of paper and place any number of points on it in any order, what is the maximum number of line segments of the same length that can be drawn between these points? Erdős himself assumed that the most advantageous arrangement of points is a strict square grid.
Solution to the Problem
The AI applied a method from algebraic number theory, building giant lattices in spaces of much higher dimensions than the usual two-dimensional plane, and then 'collapsed' these multi-dimensional figures back into two dimensions, obtaining their flat projection.