The Unit Distance Conjecture Disproved — An OpenAI Model Overturns an 80-Year-Old Erdős Conjecture

On 20 May 2026 OpenAI announced that an internal general-purpose reasoning model had disproved Erdős's unit distance conjecture. Place n points in the plane: how many pairs can be exactly distance 1 apart? For the nearly 80 years since Paul Erdős posed the question in 1946, mathematicians had believed that grid-like arrangements are essentially the best possible. The model built an infinite family of configurations with polynomially more unit distances, using tools from a distant field, algebraic number theory. On the same day Timothy Gowers and eight other mathematicians published a human-verified, digested version of the proof together with commentary.

Quanta Magazine called it "the first historically significant proof to come from an AI model." Within the 2026 wave of Erdős problems solved with AI it stood apart, because it was not an obscure problem but a famous conjecture at the centre of a field.

The unit distance problem

As OpenAI explains it, let u(n) be the largest possible number of unit-distance pairs among n points. Points on a line give n−1 pairs and a square grid about 2n, while a cleverly rescaled square grid gives n^(1+C/log log n), essentially unchanged as the best lower bound since Erdős's 1946 construction. Erdős conjectured an upper bound of n^(1+o(1)), meaning nothing could do substantially better than the grid. On the other side, the best upper bound, O(n^(4/3)) from Spencer, Szemerédi and Trotter in 1984, had also remained essentially unchanged.

OpenAI's post quotes the 2005 book Research Problems in Discrete Geometry calling it "possibly the best known (and simplest to explain) problem in combinatorial geometry", and Noga Alon describing it as "one of Erdős' favorite problems."

The construction the AI found

The model showed that for some fixed δ>0 there are, for infinitely many n, configurations of n points with at least n^(1+δ) unit-distance pairs. If the conjecture were true this would be impossible, so the conjecture is false.

In OpenAI's account, Erdős's original construction can be understood through the Gaussian integers, numbers of the form a+bi. The new argument replaces them with algebraic number fields that have richer symmetries, and uses infinite class field towers and Golod–Shafarevich theory to show the fields needed actually exist. These ideas were well known to algebraic number theorists, but their bearing on a question about the Euclidean plane came as a great surprise.

What the AI did, and what humans did

A footnote to the companion paper says the AI proof was "first mathematically generated in one shot by an internal model at OpenAI, and then expositionally refined through human interactions with Codex." According to Quanta, the author of the paper presenting the original solution was given simply as "OpenAI".

Checking fell to outside mathematicians. Daniel Litt of the University of Toronto writes that Mark Sellke and Mehtaab Sawhney at OpenAI asked him to check the solution, and that it did not take long to convince himself it was correct. The nine-author companion paper presents "a short, digested, human-verified version" of the proof and notes that its ideas may, at least in retrospect, be attributed to Ellenberg–Venkatesh, Golod–Shafarevich and Hajir–Maire–Ramakrishna. On the same day, Will Sawin of Princeton posted a paper making the exponent explicit at 1.014.

In his commentary Bloom wrote that "while the original proof produced by AI was completely valid, it was significantly improved by the human researchers at OpenAI and the many other mathematicians involved in the present paper," and stressed the vital role humans still play in discussing, digesting and improving such a proof. According to Quanta, human mathematicians substantially improved on the result within weeks, and within days related techniques were used on other important problems; a group of four mathematicians including Bloom disproved a version of the sum-product conjecture for real numbers.

A milestone, or a home advantage?

The strongest praise came from Gowers: "There is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics: if a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that."

Yet Gowers also recounted that he first misheard the news as an AI proof of the upper bound, spent the evening thinking that maybe it would soon be all over for mathematicians, and felt relief on learning it was a disproof. "Perhaps what we are seeing at the moment is not that AI is about to overtake human mathematicians, but rather that there are certain styles of problem where it has a distinct advantage," he wrote, while adding that his bet was that progress in AI mathematics is not about to plateau.

Bloom noted that most human effort had probably gone into proving the upper bound rather than trying to disprove it, and that some in the area might be disappointed that the result introduces no powerful new geometric tools. Melanie Matchett Wood of Harvard argued that the experts assembled for the paper would probably have found a counterexample had they spent similar time on it a month earlier, but that without the AI's claim nobody would have had a reason to look. She warned that "it will be easier for AI to convince humans it has a proof than to come up with a correct mathematical argument" in many cases, that mathematicians are not sufficiently prepared for this, and that closely related prior work was not appropriately referenced in the AI's paper.

What changed, and for whom

Unlike specialised systems such as AlphaProof (2024), or contest problems such as those behind gold-medal level at IMO 2025, this was a general-purpose model solving a famous research problem. The next day Google DeepMind announced AlphaProof Nexus. Quanta wrote that Erdős's problems had become a central proving ground, and in effect a series of PR coups, for the world's biggest technology companies.

For the mathematical community, citation practice, the division of checking work and the attribution of credit stopped being abstract questions. Wood called for the community to plan urgently for keeping work rigorous and correct, properly acknowledging the influence of earlier ideas, and preserving a high level of human understanding of mathematics. The story continues on the AI and mathematics timeline, and the company's history is on the OpenAI page.