“How many pairs of points can have exactly the same distance from each other on a surface?” This is the “planar unit distance” problem, for the solution of which the famous Hungarian mathematician Paul Erdős even promised prize money in 1946. However, despite this extra motivation, researchers have not been able to solve the problem in the past 80 years. They had only agreed on a conjecture; that a square grid is essentially the best arrangement to maximize the number of point pairs with the same distance.

An internal AI from OpenAI, a general reasoning model, has now refuted this assumption, according to the company: there are arrangements of points where there are even more pairs with the same distance. However, how the AI arrived at its solution is perhaps the real key. For this problem from geometry, it used another area of mathematics: algebraic number theory.

Like solving an architecture problem with music theory

Simply put, the distances between two points are always an equation. So, the problem could also be solved by searching for sets of points for which this equation is solvable unusually often. Algebraic number theory works with significantly more and more exotic number ranges than geometry. This allowed the AI to find a way for many more points on a surface to have the same distance from each other apart than previously thought. Or, expressed figuratively: Geometry has so far tried to solve the problem with Lego bricks. The AI, using algebraic number theory, has now found building blocks that are much more sophisticated and can therefore be assembled more effectively.

By the way, while this problem is very abstract, the correct arrangement of points in space is a very everyday problem. From satellites to mobile phone masts and Wi-Fi routers to navigation, they all need the most optimal arrangement possible relative to each other so that, for example, there are no dead zones but also no signals that overlap too much. If the “planar unit distance” problem were solved, it could give us new insights into the perfect arrangement of things in space that would also help us with everyday matters.

Several mathematicians have checked the AI's proof and found it to be correct. However, in their statement, the scientists point out that “the reasoning” is “decisively” based on “ideas” that “at least in retrospect, can be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.” The statement was published by OpenAI and on the preprint server arXiv.

Regardless of the specific mathematical problem and its solution, however, the fact that this AI has now worked so consistently and stringently on such a complex problem opens up new possibilities for the use of AI in mathematical research.

Will the AI also receive the prize money that Paul Erdős offered? It could urgently use it. Because while it is puzzling over mathematical problems, humanity has long since not solved the problem of its expensive electricity hunger.

(rie)