The problem was proposed by Paul Erdős, a Hungarian mathematician. It is one of the most famous questions in combinatorial geometry. The problem is stated as follows: For n points on a plane, what is the maximum number of pairs of points that have a distance of exactly one?
For many years, numerous mathematicians attempted to solve it. The best method previously conceived involved arranging points into a square lattice. However, on 20/5, OpenAI announced that its internal AI had discovered a new arrangement of points, disproving this hypothesis and generating a significantly larger number of unit distance pairs than before.
The breakthrough in the model's solution lies in using algebraic number theory to solve a geometry problem. OpenAI stated that this is a general-purpose reasoning AI, not specifically trained for mathematics. The solution was also independently verified by a group of mathematicians.
According to OpenAI, mathematicians were "stunned" by the AI's solution. Professor of Mathematics and Computer Science Noga Alon from Princeton University, said the model is an outstanding achievement and the solution is a "shock" to existing knowledge.
"I believe every mathematician in combinatorial geometry has thought about this problem. The AI built, analyzed, and applied the complex tools of algebraic number theory elegantly and skillfully", he said.
Tim Gowers, a Fields Medal-winning mathematician, called the result "a milestone in AI mathematics." Number theorist Arul Shankar said the solution proves that current AI models are not limited to assisting mathematicians but also capable of thinking and executing ideas.
Professor of Mathematics Jacob Tsimerman at the University of Toronto, said he tried to find examples to refute the AI's solution but failed.
"This is certainly a challenging work to observe closely, even if you know what is happening, and experiencing it yourself is even more difficult", he stated.
OpenAI stated that strong mathematical reasoning capabilities can transform AI into a powerful research assistant, charting new directions and helping researchers solve complex or time-consuming problems.
"If a model can perform complex reasoning, connect interdisciplinary ideas, and produce validated results, these are also incredibly useful capabilities in biology, physics, materials science, engineering, and medicine", the announcement stated.
Doan Hung (According to OpenAI)
