OpenAI Defeats Erdős on a 1946 Problem: Nine Mathematicians Sign the Verification
In recent days, OpenAI announced that its internal general reasoning model has produced a counterexample to the Erdős conjecture on unit distances in the plane, formulated in 1946. A few months ago, we had already touched upon the theme of the problems posed by mathematician Paul Erdős, a collection of open questions left behind by the brilliant 20th-century mathematician, often simple to state but extremely difficult to solve, which today represent an ideal testing ground for measuring progress, both human and artificial, in mathematical understanding.
The Unit Distance Problem
The question at the heart of the problem is simply stated, yet it conceals considerable complexity. We arrange n points on a plane: for each pair of points, we measure the distance that separates them. If the points are in a generic position, these distances are likely all different from one another. However, by placing them appropriately, we can create configurations where many pairs share the same distance: think of an equilateral triangle, where all three sides have the same length, or any regular polygon, where the number of equidistant pairs is even greater.
Erdős's question is how far one can push this game: given n points to arrange freely, what is the maximum number of pairs that can all be at the same distance from one another? By convention, that recurring distance is set equal to 1, hence the name "unit distances" of the aforementioned problem. In reality, the value is arbitrary, and what counts is the frequency with which the same distance repeats among different points in the configuration.
In particular, Erdős wondered how quickly this maximum count grows as n increases. The more points added, the more pairs become possible and the more coincidences can emerge: at what rate of growth? His 1946 conjecture was that the number of unit pairs grew slightly faster than the number of points themselves, but so little that, at large scales, the difference would essentially vanish.
Erdős's intuition held for about eighty years: the best-known concrete configuration was a square grid of points that yielded a count just above linear, enough to exclude perfectly proportional growth to the number of points, but close enough to linear to lend plausibility to his conjecture. On the opposite side of the problem, there is a theoretical ceiling on the maximum number of unit pairs, proven in 1984 by Spencer, Szemerédi, and Trotter: n^(4/3). A wide margin compared to linear that had not been questioned for over forty years.
OpenAI's internal model tackled the problem with a decidedly original approach, constructing an infinite family of arrangements of points in which the count of unit pairs grows significantly faster than linear, by a fixed amount. The difference with the square grid is qualitative: in the grid, the gain over linear was a correction that became less significant as n increased, whereas in the new construction, the gain grows with n and pulls away from pure linear by a factor that amplifies as more points are added. This is what is mathematically called a polynomial improvement, and it is exactly the kind of leap that Erdős had excluded with his conjecture, which is thus refuted.
Will Sawin, a professor at Princeton and one of the signers of the verification paper, subsequently refined the construction and derived an explicit exponent of 1.014: the count of unit pairs grows as the number of points raised to the 1.014 power. To give an order of magnitude, with a million points the count is about 21% higher than the strictly linear case; with a billion, over 30% higher; the gap continues to widen as n increases. The original proof of the model gave a much smaller exponent, equal to 1 plus a quantity on the order of 10^-38: numerically negligible, but sufficient as a formal counterexample to the conjecture.
What struck mathematicians was the approach taken by the model, which instead of starting from discrete geometry, i.e., the field in which the problem has been studied for decades, chose to interpret it under the algebraic number theory. This is a branch of mathematics that covers prime numbers and arithmetic properties of integers, rather than points on a plane, and to which discrete geometry specialists rarely turn when addressing problems like this. The real value in the result of OpenAI's model lies more in the connection between two distant areas of mathematics than in the solution of the problem itself.
The Verification of the Nine and Precautions
The verification paper is signed by Noga Alon, Thomas Bloom, William T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood. Gowers, a Fields Medal winner, writes that if a human had submitted this paper to the Annals of Mathematics and a quick evaluation had been requested, he would have recommended acceptance without hesitation. It is one of the strongest public judgments expressed thus far on a mathematical result generated by AI.
Bloom, curator of the website www.erdosproblems.com, also made a particularly relevant observation: the solution found by OpenAI's model is, when assessed retrospectively, a natural generalization of Erdős's original based on lattices, and much of the community had not attempted to refute the conjecture because they were convinced it was true. The AI succeeded primarily because it explored a hypothesis that experts had discarded. Melanie Matchett Wood notes that the nine mathematicians who verified the result would likely have been able to find a counterexample to Erdős's conjecture themselves if directed to do so, but the point is that this group would probably never have spontaneously formed: without the AI's output, no one would have gathered the group of experts who then verified the result.
In closing, a clarification highlighted by OpenAI itself: the proof was "generated in a single pass by an OpenAI internal model and then refined expositorily through human interactions with Codex." In other words, the mathematical idea is from the model, but the final form is the result of collaborative work.
The counterexample does not affect the best known upper limit, n^(4/3): the gap between the minimum number of unit pairs that can be guaranteed and the maximum that can be excluded remains wide, and the new construction narrows it only slightly. The exponent 1.014 demonstrates that Erdős's conjecture is false, but does not resolve the doubt about what the correct answer is. To date, there appears to be no formal submission of the paper to a peer-reviewed scientific journal, despite Gowers' favorable judgment for the Annals of Mathematics.
Finally, the relevance of OpenAI's announcement should also be read in the context of a broader framing: in October 2025, former OpenAI Vice President Kevin Weil shared on X that GPT-5 had solved ten previously open problems of Erdős and made progress on another eleven. The responses from Yann LeCun and Demis Hassabis of Google DeepMind forced Weil to delete the post, after Bloom himself called the claim "a dramatic misrepresentation" since the model had solved nothing and had merely retrieved solutions already present in literature.