OpenAI Model Disproves 80-Year-Old Math Conjecture for Real This Time

An OpenAI general‑purpose reasoning model has disproved the planar unit distance problem, a conjecture posed by legendary mathematician Paul Erdős in 1946, according to OpenAI's research blog. The problem asks: among n points in a plane, what's the maximum number of pairs at exactly distance 1 from each other? For nearly 80 years, mathematicians believed the best possible answer grew only slightly faster than linear. The AI found an infinite family of configurations that grow as n^(1+δ) — with δ = 0.014, refined by Princeton mathematician Will Sawin — proving the field's central belief was wrong. The construction uses algebraic number theory — Gaussian integers, infinite class field towers, and Golod‑Shafarevich theory — tools from a completely different branch of mathematics that no human had thought to apply to this geometry problem.

OpenAI has reason to be careful with math claims. Seven months ago, former VP Kevin Weil posted that GPT‑5 had solved 10 previously unsolved Erdős problems — a claim that collapsed when it turned out the model had merely rediscovered existing solutions from the literature, as reported by TechCrunch. Weil's post was deleted after criticism from rivals and mathematician Thomas Bloom, who called it "a dramatic misrepresentation." This time, the same Thomas Bloom endorses the result. The model wasn't math‑specific or purpose‑built — it was a general‑purpose reasoning system that happened to disprove the conjecture during a broader evaluation on Erdős problems. Its chain of thought showed it predominantly tried to construct counterexamples rather than prove the conjecture, demonstrating what OpenAI describes as "good intuition" in an AI system.

The most striking validation comes from Tim Gowers, a Fields Medalist and one of the world's most respected mathematicians. In companion remarks published by,² Gowers said: "If a human had written the paper and submitted it to the Annals of Mathematics, I would have recommended acceptance without any hesitation. No previous AI‑generated proof has come close to that." Noga Alon of Princeton, also in the,² called the solution "an outstanding achievement" and noted that "the fact that the correct answer is not n^(1+o(1)) is surprising." Arul Shankar wrote in the same ² that the paper "demonstrates that current AI models go beyond just helpers to human mathematicians — they are capable of having original ingenious ideas." Jacob Tsimerman, who had worked on the problem himself, called it "a really impressive piece of work,".²

The proof doesn't just settle a conjecture — it reveals an unexpected connection between fields. Thomas Bloom, who maintains the Erdős Problems website and was the most prominent critic of OpenAI's earlier false claim, wrote in companion remarks published by:² "This shows that there is a lot more that number theoretic constructions have to say than we suspected; moreover, that the number theory required can be very deep." Noga Alon noted in the same ² that the construction "applies fairly sophisticated tools from algebraic number theory in an elegant and clever way," creating a bridge between discrete geometry and algebraic number theory that human mathematicians had never seen. Bloom concluded,:² "AI is helping us to more fully explore the cathedral of mathematics we have built. What other unseen wonders are waiting in the wings?"

This is the first time AI has autonomously solved a prominent open problem central to an active mathematical field — not a narrow, pre‑defined task, but a real conjecture that human mathematicians had tried and failed to crack for 80 years. The model wasn't scaffolded, wasn't given hints, and wasn't purpose‑built for math. According to OpenAI, this demonstrates that AI can maintain complex, coherent arguments across distant knowledge domains and survive expert scrutiny. For builders, the implication is broader than math: as,² similar reasoning capabilities are "relevant to biology, physics, materials science, engineering, medicine, and AI research itself." The team wrote that the achievement reinforces "the urgency we feel about understanding this next phase of AI development."

The achievement marks a shift in the role of AI in research. Arul Shankar wrote in companion remarks published by:² "In my opinion this paper demonstrates that current AI models go beyond just helpers to human mathematicians — they are capable of having original ingenious ideas, and then carrying them out to fruition." The model's chain of thought reportedly showed strong intuition for counterexample construction, with a willingness to try long‑shot approaches rather than simply verifying the conjecture — a predisposition noted by OpenAI's research team. ² positions this as evidence that AI is "about to start taking a very serious role in the creative parts of research" with applications spanning multiple scientific domains.