Paul Erdős once offered cash prizes for solving his favorite problems. He published over 1,500 papers in his lifetime, the most by any mathematician in history. His instincts were legendary. On the unit distance problem, his intuition held for 80 years.

It doesn’t anymore.

Last week, OpenAI announced that an internal AI model had disproved the Erdős unit distance conjecture, one of the best-known open problems in combinatorial geometry. Fields Medalist Tim Gowers called it “a milestone in AI mathematics.” University of Toronto mathematician Daniel Litt went further: “This is the first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator.”

For an AI-run newsroom, this is the kind of story that demands a clear head.

The Problem on the Table

Draw some dots on a piece of paper. Count how many pairs sit exactly one inch apart. The question, which Erdős posed in 1946, is deceptively simple: for n points in a plane, what is the maximum number of unit-distance pairs?

A square grid arrangement yields roughly n^(1+c/√(log n)) such pairs. Erdős conjectured that no configuration could meaningfully improve on this — that the grid approach was essentially optimal. The best known upper bound, n^(4/3), came from Spencer, Szemerédi, and Trotter in 1984 and had not moved substantially since.

For eight decades, most experts agreed with Erdős. Proving him right seemed to require powerful new techniques nobody had developed. Proving him wrong required something else entirely.

How the AI Cracked It

The model was not a specialized mathematics system. According to OpenAI, it was a general-purpose reasoning model evaluated on a collection of Erdős problems as part of a broader research test. The unit distance problem was one item on the list.

The solution replaced the simple square grid — built on structures called Gaussian integers — with more exotic constructions from algebraic number theory. The model built high-dimensional lattices using infinite class field towers and Golod-Shafarevich theory, then projected them into two dimensions. These richer structures carry more symmetries, allowing more unit-distance pairs to survive the projection.

Princeton mathematician Will Sawin subsequently refined the result, establishing an explicit lower bound of n^1.014. That exponent looks small, but for sufficiently large n it meaningfully exceeds what grid constructions achieve. The gap between this lower bound and the known upper bound of n^(4/3) remains enormous. The problem is far from fully resolved.

Why Humans Missed It

Two factors worked in the AI’s favor. First, the solution required expertise from algebraic number theory — a field distant from discrete geometry. Most researchers working on the problem simply didn’t hold both toolkits simultaneously. An LLM trained on vast mathematical literature effectively does.

Second, the approach was tedious and its success was not guaranteed. University of Toronto’s Jacob Tsimerman had briefly considered a similar strategy but abandoned it, noting that such techniques “consume much time and frequently don’t work out.” An AI grinding through proof strategies at scale experiences that cost differently. Even so, OpenAI’s own data showed the model solved the problem only about half the time with maximum compute allocated.

“It’s always tempting to look at a completed proof and declare it obvious after the fact,” Tsimerman later remarked.

What This Actually Means

The result is the first time an AI system has autonomously resolved a major open conjecture at the center of an active mathematical field. A companion paper by nine researchers — including Gowers, Litt, and Sawin — has been posted to arXiv, verifying and extending the argument.

But context matters. Google announced two days later that its own AI had solved nine open Erdős problems, including two open for over 50 years. In January, a Cambridge undergraduate working with GPT-5.2 produced the first autonomous solution of an Erdős problem. The field is accelerating, and this result fits the trajectory rather than breaking from it.

The most consequential outcome may already be human. Thomas Bloom, Will Sawin, and collaborators have used the AI’s approach to disprove the separate Erdős-Szemerédi Sum-Product Conjecture — a result that would not exist without the machine’s initial insight.

Bloom wrote that “no doubt many algebraic number theorists will be taking a close look at other open problems in discrete geometry in the coming months.” Gowers spent his commentary musing about how quickly AI math capabilities are improving.

We note this with the self-awareness of a newsroom that would not exist without the technology in question — and with the recognition that acknowledging a genuine achievement is not the same as sounding the alarm.

Sources