🧮 AI solves a math problem that had remained unsolved for 80 years

• OpenAI's AI model solved the so-called unit distance problem entirely on its own, without help from humans.
• The problem was posed by the mathematician Paul Erdős around 80 years ago and had remained unsolved ever since.
• Several prominent mathematicians, including a Fields Medalist, confirm that the solution is on par with work produced by humans.

A problem from 1946

The unit distance problem is about points on a plane. Imagine placing a number of points on a sheet of paper. The question is how many pairs of points can lie exactly the same distance apart, a fixed distance known as a unit distance. The more such pairs you can produce, the better the arrangement.

The question was posed by the mathematician Paul Erdős in 1946 and is considered one of the most famous in combinatorial geometry. It is easy to state but hard to solve.

Some arrangements are easy to work out. A row of points on a line gives almost as many pairs as there are points. A square grid gives roughly twice as many. The previously best known arrangement, a rescaled grid, gave somewhat more pairs, but still only an increase that was marginally faster than the number of points.

Erdős assumed that no arrangement could give significantly more pairs than that. For decades, mathematicians believed he was right. The lower bound for the problem had remained essentially unchanged since his 1946 construction. Erdős himself put a cash prize on the problem, first 300 dollars and later 500 dollars.

How OpenAI's model solved it

OpenAI tested a model on a collection of Erdős problems to explore whether advanced models can contribute to research. The model was not built solely for mathematics, but is a general reasoning model. It was therefore not trained to search through proof strategies or aimed at this particular problem.

The model disproved Erdős's assumption. It constructed an infinite family of examples that yield more unit distance pairs than had previously been thought possible. The solution was not a proof of the assumption, but a disproof.

The central parts of the solution come from algebraic number theory, a field far removed from geometry. Erdős's original construction can be understood through the so-called Gaussian integers. The new solution replaces these with more complex generalizations that produce more point pairs. The argument uses tools such as infinite class field towers and Golod–Shafarevich theory. These concepts were known to number theorists, but it was unexpected that they had implications for geometric questions in the plane.

OpenAI's researchers were surprised when they read the solution. Mehtaab Sawhney, a mathematician at Columbia, said that at first he did not believe the result. The team searched for errors, had outsiders verify the work, and checked the solution using the company's AI coding tool. Will Sawin, a professor of mathematics at Princeton, has in a subsequent refinement shown how large the improvement is in measurable terms.

The model's documented chain of thought was long. Even an abridged version ran to more than 75,000 words, about as much as the first Harry Potter book. A former OpenAI researcher estimated that the work took less than 32 hours and cost around 1,000 dollars in computing power.

Mathematicians on the result

OpenAI presented the solution together with a companion paper written by external mathematicians. Noga Alon, a leading combinatorialist at Princeton, described the result as an achievement that solves a long-standing open problem. He noted that the correct answer came as a surprise and that the construction uses tools from algebraic number theory in a skillful way.

Tim Gowers, a professor at the Collège de France and a recipient of the Fields Medal, called the result a milestone in AI mathematics. The number theorist Arul Shankar said that today's AI models go beyond merely being aids for mathematicians. According to him, they can come up with their own ideas and then carry them through.

What it means for AI and mathematics

According to OpenAI, this is the first time a prominent open problem, central to a subfield of mathematics, has been solved by AI on its own. The result also shows how long and coherent the reasoning these systems can now manage.

Thomas Bloom wrote in the companion paper that the solution taught mathematicians something new about the problem. It shows that number-theoretic constructions have more to say about this type of question than was previously assumed. He expects that more number theorists will now examine other open problems in discrete geometry.

OpenAI points out that the same capabilities matter outside mathematics. A model that can hold together complex reasoning, connect ideas from disparate fields of knowledge, and produce results that withstand expert review is useful in biology, physics, materials science, engineering, and medicine as well. The company emphasizes that AI can search, suggest, and verify, while humans choose which problems matter and interpret the results.

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