Some mathematical problems are formulated so simply that they can be explained to a child, yet so difficult that they occupy generations of mathematicians. One such problem is the so-called unit distance problem: Place \(n\) points in the plane. How many pairs of points can then have a distance of exactly \(1\) ? The problem dates back to Paul Erdős and has been studied since 1946. OpenAI has now published that an internal model has disproven a long-held conjecture about this problem.
At first glance, the question sounds harmless. If you place \(n\) points on a straight line, you get approximately \(n-1\) intervals of length \(1\) . If you arrange the points as a grid, like on graph paper, you get significantly more of these intervals: horizontally and vertically between adjacent points. Erdős had already found constructions that were somewhat better than linear. For a long time, however, it was believed that this could not be significantly improved. Formally, the assumption was that the maximum number of such unit intervals only grows approximately \(n^{1+o(1)}\) , that is, somewhat faster than \(n\) , but not with a fixed additional exponent.
This is precisely the surprising point: The OpenAI model (which one, was not disclosed) constructed not just a single counterexample, but an infinite family of point sets \(P\) for which the number of unit distances is at least \(|P|^{1+\delta}\) , with a fixed \(\delta>0\) . A later refinement by Will Sawin, according to OpenAI, even yields \(\delta=0{,}014\) . This sounds small, but is mathematically enormous: It is no longer a logarithmic remainder, but a true polynomial gain.
A simple example illustrates the basic idea. Let's consider the complex number
\[
u=\frac{2+i}{2-i}=\frac{3+4i}{5}=\frac35+\frac45i.
\]
This number has an absolute value of \(1\) , because
\[
\left|\frac35+\frac45i\right|=\sqrt{\left(\frac35\right)^2+\left(\frac45\right)^2}=1.
\]
Therefore, if \(x\) is a point in the plane, then \(x\) and \(x+u\) are exactly \(1\) apart. Specifically, for example, the points are located at...
\[
0
\quad\text{und}\quad
\frac35+\frac45i
\]
exactly one unit apart. A number-theoretic expression thus yields a geometric direction of length \(1\) . This is not yet the grand counter-proof, but it is the small version of the trick: one looks not only for horizontal and vertical unit distances as in the grid, but for many arithmetically generated directions, all of which have exactly length \(1\) .
The actual construction uses significantly more powerful tools. Instead of working only with the Gaussian integers \(\mathbb{Z}[i]\) , the proof uses more complicated algebraic number fields \(K=L(i)\) . There, many elements of the form
\[
u=\frac{\alpha}{c(\alpha)}
\]
constructed, where \(c\) plays the role of complex conjugation. The crucial effect is: among the relevant complex embeddings, these \(u\) each have a magnitude of \(1\) . They are therefore candidates for many different unit directions.
In very rough terms, a true counterexample doesn't look like a small, pretty picture with ten dots, but rather like a huge, arithmetically constructed swarm of dots. You take a high-dimensional grid, cut out suitable points from it, and then project these back onto the ordinary plane. In proof notation, this is expressed roughly in the form...
\[
P_j=\pi_1\big((y+\Lambda_j)\cap W\big),
\]
In other words: One takes points from a shifted grid \(y+\Lambda_j\) , restricts them by a region \(W\) , and projects them with \(\pi_1\) onto a complex coordinate, i.e., onto \(\mathbb{C}\cong\mathbb{R}^2\) . Many differences between these points are then precisely such \(u\) elements with magnitude \(1\) . Therefore, after the projection, they become true unit distances in the plane.
A visual analogy is this: The standard grid uses a few simple directions, such as right, left, up, and down. The new construction, however, generates a vast number of hidden directions derived from algebraic number theory. Each direction is exactly one unit long. Because there are so many such directions and many points correspond to them, the total unit distances are greater than the old conjecture would allow.
It is also noteworthy where the proof comes from. According to OpenAI, the solution was found autonomously by a general reasoning model, not by a mathematical system specifically trained for this problem. The proof was then reviewed internally and externally and rendered into a human-readable form. The accompanying notes from external mathematicians also emphasize that this is not just an automated version of a known method, but an unexpected connection between discrete geometry and algebraic number theory.
This is probably the real reason why this result is so interesting. It's not just about an AI solving a problem correctly. It's about it finding a way that wasn't obvious to many people: tackling a geometric problem about distances in a plane with deep tools from number theory. This doesn't make mathematics any less human. But it does give it a new, rather uncomfortably powerful opponent.