On 20 May 2026, OpenAI announced that an internal reasoning model had disproved the Erdős unit distance conjecture, a question about points in the plane that Paul Erdős posed in 1946. Mathematicians outside the company checked the argument, and several have called it significant. It is also an informal proof from an unreleased model, with the details of the run left undisclosed, so the right reading is “important and credible, not yet fully absorbed.”

Here is what the Erdős unit distance conjecture actually says, what the new construction does, who has checked it, and where the open questions sit.

The problem in one picture

Scatter n points on a sheet of paper. Count the pairs of points that are exactly one unit apart. What is the largest that count can be, as n grows? Call the answer u(n).

Small cases are easy to play with. A triangle of three points gives three unit pairs. A honeycomb-like arrangement packs in more. The hard part is the long-run growth rate. Erdős conjectured in 1946 that u(n) grows only slightly faster than n itself, which OpenAI’s write-up states as an upper bound of n^(1+o(1)). In words: any power of n above 1, however close to 1, would eventually be too big.

What was known before

Two facts framed the problem for decades. On the construction side, the best known arrangements came from a rescaled square grid, giving about n^(1+C/log log n) unit pairs. That grows barely faster than a straight line, which is why the conjecture looked plausible. On the bound side, the best proven upper limit is O(n^(4/3)), a 1984 result of Spencer, Szemerédi and Trotter that has stood essentially unchanged since.

Between “just above n” and “n to the 4/3” sits a wide gap. The Erdős conjecture bet on the bottom of it. OpenAI’s page also notes partial supporting evidence from related work in non-Euclidean settings, by Matoušek and by Alon, Bucić and Sauermann, which made the conjecture look more credible, not less.

What the model’s construction does

The announcement describes a proof that there is an infinite family of point sets with at least n^(1+δ) unit-distance pairs, for some fixed δ greater than zero. That single sentence is enough to kill the conjecture, because the conjecture said no such δ could exist.

The ingredients, as OpenAI summarizes them, are number-theoretic. The grid construction effectively lives in the Gaussian integers, the complex numbers whose real and imaginary parts are whole numbers. The new argument replaces them with more elaborate algebraic number fields that have richer symmetry, which allows many more points to sit at mutual distance one. To show that fields with the needed properties exist, the proof invokes infinite class field towers and Golod-Shafarevich theory, both from algebraic number theory rather than from geometry.

That crossover is part of why mathematicians took notice. A problem that looks like pure plane geometry was cracked using tools from a distant corner of the subject.

Who checked it

OpenAI says a group of outside mathematicians verified the proof and wrote a companion paper explaining the argument and its background. The page quotes Tim Gowers, who calls the result “a milestone in AI mathematics,” and also quotes Noga Alon, Arul Shankar and Jacob Tsimerman. Thomas Bloom, who maintains the Erdős problems site, wrote the companion note and described the answer to whether we understand the problem better now as a “moderated yes.”

A second thread came from Princeton. Will Sawin read the proof, then wrote a paper that went step by step through the argument to make it more efficient and to extract an explicit exponent. In an interview published by Gizmodo on 1 June, Sawin said the original argument was not wrong and had achieved OpenAI’s goal; his changes served a different goal. His version gives an exponent of a little more than 1.01. The same article reports that other researchers have since pushed it to about 1.03. OpenAI’s page mentions a refinement reaching δ = 0.014, which matches that “a little more than 1.01” figure.

Version What it gives
Erdős conjecture (1946) u(n) is at most n^(1+o(1))
Grid construction (prior best) About n^(1+C/log log n)
OpenAI model proof (May 2026) At least n^(1+δ) for some unspecified δ above 0
Sawin refinement Explicit exponent a little over 1.01
Later improvements (per Gizmodo) About 1.03
Proven upper bound (1984) O(n^(4/3)), about 1.33

What the result does not show

Sawin’s own assessment is the most useful corrective. He told Gizmodo that the central idea, using number fields of growing degree, is not something humans could never have found, and he placed the result “somewhere in between” the hype and the dismissal. The proof is informal, though in principle it could be formalized in a proof assistant. It is also asymptotic: it says such configurations exist for large n without handing over a concrete point set for any particular n.

There are limits on what outsiders can evaluate. The model is internal and unreleased. OpenAI says it measured success rates at different amounts of test-time compute and shows this in a figure, but the text reports no numbers, and Gizmodo notes the company did not describe the computing power used, how many problems were tried, or how the system was set up. That matters for judging how repeatable the feat is.

The claim that this is the first prominent open problem solved autonomously by AI is OpenAI’s own framing. It deserves the usual treatment given to a company’s description of its own result, even when outside experts are supportive.

One more caution concerns the word “autonomously.” OpenAI says the model produced the proof, and outside mathematicians then took over the checking, the explanation and the refinement. Humans framed the question, selected it as worth attention, and interpreted the output. That division of labor is how the result reached the literature, and it is worth keeping in mind when headlines compress the story to a machine solving a problem alone.

Why this one is different from earlier AI-math claims

Earlier announcements of AI “solving” open problems have sometimes turned out to be rediscoveries of results already in the literature. This case differs in the obvious way: the conjecture was a named, 80-year-old problem, the answer contradicts what many experts expected, and independent mathematicians produced a companion paper and a refinement within days to weeks. A rediscovered result does not behave that way.

The same questions apply to every future claim of this type: is the problem named and old, do independent experts engage in detail, and does the argument hold up when others rework it?

Why the exponent matters

An exponent like 1.01 sounds like a rounding error next to 1. For small n it is. Take n equal to a million: n^1.01 is about 1.15 million, only 15 percent above a straight line. But the conjecture was a statement about all large n, and growth rates compound. The difference between “slower than every power above 1” and “faster than n^1.01” is the difference between a ceiling that never lets the count escape linear growth and one that does. The distinction is qualitative, not a matter of degree, and that is why a tiny δ settles a question that stood for eight decades, even though no one could draw the resulting point sets by hand.

Likewise, the gap that remains is large. Between roughly 1.03 and 4/3, which is about 1.33, lies a range where the truth might sit. Closing it from either side would be a major result in its own right.

How a problem like this gets absorbed

Mathematics has a slow, social process for new proofs. First, specialists read the argument to see whether any step fails. Then others rewrite it more cleanly, as Sawin did, which tends to expose both the core idea and the slack. Later come simplifications, extensions to related problems, and eventually a textbook treatment. The companion paper written by outside mathematicians is an early step in that chain.

This is also why the informal status of the proof is less alarming than it sounds. Most human proofs begin as informal arguments in papers, and a formal check in software is a bonus rather than a prerequisite for the community to accept them. What is unusual here is the source, not the format.

What to watch next

  • A formal, machine-checked version of the proof, which would remove doubt about subtle gaps.
  • Further improvements to the exponent, which would show how loose the first construction was.
  • Peer-reviewed publication of the companion material.
  • Any release of the model, or a description of how many attempts were needed.

The gap between 1.03 and 4/3 remains. Nobody yet knows where the true growth rate of u(n) lies, and the new construction only shows that the bottom of the range was the wrong place to look.

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