Yes—an OpenAI model produced a proof that contradicts Erdős’s conjecture about how many pairs of points in the plane can be exactly one unit apart. External mathematicians then checked the argument and wrote a human-digested exposition. It is a notable AI-generated result in a specific area of mathematics, not evidence that AI can solve mathematics broadly or reliably.
What problem did the AI solve?
The question is simple to state: given n points in the Euclidean plane, what is the greatest possible number of pairs that are exactly distance 1 apart? This is known as the planar unit-distance problem, a longstanding question in combinatorial geometry. OpenAI’s account describes it as one of Erdős’s favorite problems; mathematician Noga Alon recalled hearing Erdős mention it in lectures. OpenAI’s announcement gives the historical context.
Erdős conjectured that the maximum number of such pairs grows no faster than n1+o(1). In plain language, the number could grow a little faster than linearly, but its exponent would approach 1 as the point set gets larger. The model’s construction instead gives at least n1+δ unit-distance pairs for infinitely many values of n, where δ is a fixed positive number. That polynomial improvement over linear growth contradicts the conjecture.
How significant is the improvement?
The result is best understood alongside the previous bounds. Rescaled square-grid constructions were already known to produce slightly more than a linear number of pairs, with growth on the order of n1+C/log log n for a constant C. The best known upper bound cited in OpenAI’s explanation is O(n4/3), from work by Spencer, Szemerédi, and Trotter in 1984. The new result is a stronger lower bound: it shows that the maximum exceeds the near-linear behavior Erdős conjectured for infinitely many sizes of point sets. It does not determine the exact maximum.
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- Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth, paperback
| Result | What it says | Attribution |
|---|---|---|
| Erdős’s conjecture | Maximum grows as n1+o(1) | Erdős conjecture, as summarized in OpenAI’s explanation |
| Earlier lower bound | Square-grid constructions reach order n1+C/log log n, for a constant C | Prior constructions, as summarized in OpenAI’s explanation |
| Best cited upper bound | O(n4/3) | Spencer, Szemerédi, and Trotter, 1984, as cited by OpenAI |
| AI-generated construction | At least n1+δ pairs for infinitely many n; the original proof did not specify an explicit δ | OpenAI’s announcement and explanation |
| Refined exponent | δ = 0.014 | Princeton mathematician Will Sawin’s refinement, not the original generated proof, as reported by OpenAI |
Why number theory enters a geometry problem
The counterexample does not simply improve a familiar grid. Its construction uses algebraic number theory: it builds many algebraic numbers of magnitude one in number fields and uses them as differences between points. Suitable number fields can be arranged in increasingly large layers, including infinite class field towers of Golod–Shafarevich type. Those structures supply many unit-length differences, and therefore many pairs at distance one.
The older square-grid approach can be viewed as a special case involving Gaussian integers. The new construction uses richer number-field structures and symmetries to go further. That is the conceptual connection; the full argument is technical. The mathematicians’ companion exposition presents a human-digested, somewhat simplified and generalized version of the model’s proof. Read the companion paper, “Remarks on the Disproof of the Unit Distance Conjecture.”
What does “AI solved it” mean?
The model generated the core argument
OpenAI says the proof came from an internal general-purpose reasoning model, rather than a system trained specifically for mathematics or built for this particular problem. The companion paper says the mathematical argument was generated in one shot; people later used Codex interactions to refine its exposition. That distinction matters: the reported achievement is not merely that AI helped a mathematician polish a proof.
Mathematicians checked and explained it
A group of external mathematicians checked the proof and prepared the human-readable exposition. Tim Gowers, a mathematician and Fields Medalist, called it “a milestone in AI mathematics” and said he would have recommended acceptance without hesitation if asked for a quick opinion on a human-authored paper. That is a strong expert assessment, not a report that the work has been accepted by a journal. The cited accounts establish external expert scrutiny and a human-verified exposition; they do not establish journal publication or Lean verification.
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The authors also treated the result as mathematically informative, not just a striking demonstration. In the companion note, Thomas F. Bloom said the proof taught us something new about the problem, but described that assessment as “a moderated yes.” That is a useful calibration: the result is important, while its wider mathematical consequences remain a matter of judgment.
How this differs from AI solving Olympiad problems
Contest problems and open research conjectures are different tests. An Olympiad has a fixed set of questions, a known time limit, and established scoring rules. A research problem may be open for decades, with no prescribed route to a solution. The 2024 International Mathematical Olympiad result was impressive, but it should not be conflated with the unit-distance proof.
| Demonstration | Reported result | What the figure represents |
|---|---|---|
| AlphaProof and AlphaGeometry 2 at the 2024 IMO | Four of six problems; 28 of 42 points | Google DeepMind’s combined systems achieved a score equivalent to a silver medal on that year’s contest; solutions were scored under IMO rules by prominent mathematicians. Google DeepMind’s account and the 2025 Nature paper describe the work. |
| Open Erdős problems benchmark | Two of 68 selected problems | A pre-release GPT-6 Astra resolved two in the evaluation reported by Adamczewski and Bloom; four other evaluated models resolved none. The benchmark allotted $300 per problem and 72 hours of working time. The September 6, 2026 preprint reports the setup. |
The Erdős benchmark is a useful counterweight to headline-making successes, but it does not directly test the unit-distance proof. Its authors caution that celebrated demonstrations are not yet a systematic picture of AI mathematical ability, pointing to reporting bias, compute disclosure, human scaffolding, and possible contamination as concerns.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What this result does—and does not—show
- It does show that an AI model generated a substantive counterexample argument for a famous open problem in discrete geometry, and that external mathematicians scrutinized and explained the proof.
- It does not show that AI has solved mathematics as a whole, that the unit-distance problem’s exact maximum is now known, or that models will reliably solve other research problems.
- It is not a solution to a Millennium Prize problem. “One of mathematics’ biggest problems” is headline shorthand for a prominent, longstanding question in combinatorial geometry—not a formal ranking across all mathematics.
For readers who want more background on the problem’s place in the field, OpenAI’s announcement points to the 2005 book Research Problems in Discrete Geometry by Brass, Moser, and Pach, which calls it “possibly the best known (and simplest to explain) problem in combinatorial geometry.”
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