AI Model Solves 80-Year Old Erdos Problem - And Human Mathematicians Freak Out

 

   The Open AI research team that developed the model to solve unit distance problem (WSJ))

                        Fig. 1. Erdos unit distance problem before AI solution

                  Fig. 2. One of infinite no. of AI solutions to problem, actually a disproof
          

In late May, OpenAI, the maker of ChatGPT, announced that one of its models had disproved a notable 80-year-old mathematics conjecture in the field of combinatorial geometry. This was the famous unit distance problem (see Fig. 1) recently brought to the fore in a WSJ Exchange piece ('A Math Problem Stumped Everyone for 80 Years- AI Just Cracked It', May 30-31, p. B2).  According to the article: 

"The simplest form of the unit distance problem goes something like this: If you put n dots on a sheet of paper, how many pairs of dots can be exactly one unit apart?"

Then noting: 

"Erdos' showed in 1946 that arranging those dots in a grid produced a certain number of pairs (See Fig. 1) and his conjecture was that no arrangement could do much better.'

In other words, his conjecture was that the actual optimum was much closer to the lower bound than the upper one. He predicted, but couldn’t prove, that the maximum number of unit-distance pairs grows just barely faster than the number of points.

To be more precise, Erdos conjectured that the number of unit distances would be n^(1+o(1)). In other words, for a sufficiently large n, the maximum number of unit distances would be less than n^(1+𝜖) for any 𝜖 > 0. That could end up growing a little faster than his lower-bound construction—which was n^(1 + C/(log log n)) for some constant C—but within the same general ballpark.

Open AI’s research team (See image above) solved Paul Erdos's 1946 unit distance problem not by proving it, but by disproving it.  As the WSJ piece notes:

"Only by defying conventional wisdom and experimenting with seeming improbable strategies did the model find an unexpected path forward."

Why not the humans? The answer almost seems obvious:

"Humans specialize while AI synthesizes. While humans tend to focus on their specific areas of expertise, AI models use their vast knowledge of spot connections that human mathematicians seldom see themselves. In this case that meant pulling from both algebraic number theory and discrete geometry, which have about as much in common as the marathon and pole vault. In addition AI has time, attention, patience and the focus and persistence to stick with methods humans might abandon. All attributes the Erdos problem demanded."

So what did the research team's model actually do that so flummoxed so many human mathematicians?  First,their internal AI model constructed an arrangement of points in a 2D plane that yields significantly more unit-distance pairs than the linear growth rate Erdos conjectured.

The AI achieved this by taking a completely different approach from standard human intuition, which had historically focused on standard, repetitive square grids.  First, the team's AI model constructed an elaborate grid structure in a higher mathematical dimension. This high-dimensional grid was designed with special mathematical symmetries that allowed for a much higher density of equal-distance pairs.

Second, the AI model then mapped this higher dimensional grid onto the standard 2D plane, producing a flattened "shadow". Incredibly - and to human mathematicians' astonishment - instead of using standard (discrete) geometry, the AI utilized ideas from algebraic number theory (such as class field theory) to map the distances.

This counterexample proved that the maximum number of unit distances grows 'polynomially' faster than a near-linear rate for infinitely many values of n.  (A diagram from one of the AI’s disproofs of the unit distance conjecture, is shown in Fig. 2.)  Thus in May 2026, an OpenAI research team announced that its model disproved the longstanding Erdos conjecture. The model actually discovered an infinite family of configurations that beat the Erdos bound by utilizing advanced algebraic number theory.

Mathematician Will Sawin subsequently made this breakthrough explicit by establishing that the number of unit distances grows as n^{1+d }) (for some d > 0)

Though the problem sounds recreational, solving it requires deep concepts from number theory, graph theory, and algebraic geometry. The recent disproof is highly regarded in the mathematics community as a milestone achievement for autonomously generated AI mathematical research.  But on that basis it has also engendered worry from some human math quarters. More on that in a  future blog post.