The AI-led successes in math could also spur human mathematicians to new levels of achievement.
Days after OpenAI announced that one of its internal reasoning models had cracked the planar unit distance problem — an 80-year-old open question posed by the prolific Hungarian mathematician Paul Erdős — a Princeton professor has already sharpened the result, by hand, with a cleaner method and a better number.
Will Sawin, a professor at Princeton whose work spans number theory, combinatorics, and algebraic geometry, posted a paper to arXiv on May 20 showing that there exist sets of n points in the plane with at least n^1.014 unit-distance pairs. The OpenAI model had proved the same qualitative claim — a polynomial improvement over the longstanding best bound — but without pinning down an explicit exponent. A companion paper by a team of human mathematicians had since extracted a value of roughly δ ≈ 6 × 10⁻³⁸. Sawin’s exponent of 0.014 is roughly 10³⁵ times larger.
What The Problem Actually Says
The unit distance problem asks: given n points in the plane, what is the maximum number of pairs that are exactly distance 1 apart? Erdős first posed it in 1946, attached prize money to it, and conjectured that the answer was essentially linear — no construction could yield dramatically more than n unit-distance pairs. For nearly eight decades, the best known construction barely improved on that, giving roughly n^(1 + C/log log n) pairs — linear growth with a correction term so slow it barely registers.
OpenAI’s model overturned that conjecture by showing the true answer is at least n^(1+δ) for some fixed positive δ — a polynomial gap above linear, a qualitatively different regime. Fields Medalist Tim Gowers described the proof as “a milestone in AI mathematics.”
What Sawin Did Differently
The OpenAI model’s proof worked by borrowing tools from an entirely different branch of mathematics — the study of algebraic number fields, structures that generalize ordinary arithmetic — and using them to build point arrangements with surprisingly many unit distances. It was a stunning conceptual leap, but the resulting exponent was tiny and left largely implicit. Sawin took the same core idea and made every step of it more efficient. Think of it like a recipe that was proven to work, but where the ingredients were expensive and the quantities vague. Sawin rewrote the recipe using cheaper, more common ingredients — small prime numbers like 2, 3, and 5 rather than large, exotic ones — and specified exact amounts, yielding a dramatically better result from essentially the same kitchen.
The improvements are technical but the principle is simple: the OpenAI proof worked under unnecessarily restrictive assumptions, which forced it to work with large, hard-to-use numbers. Sawin dropped several of those constraints at once, each time finding a tighter tool that did the same job with less waste. The final exponent of 0.014 — while not the end of the story — is not a marginal refinement. It is a result that is actually explicit and large enough to be meaningful. Tellingly, Sawin’s paper is about the same length as the original OpenAI write-up, which suggests the gains came from thinking more clearly about the problem, not from doing more work.
Part Of A Broader Pattern
This is not an isolated dynamic. AI models have been racking up mathematical milestones in recent months. GPT-5.2 helped solve Erdős Problem #281, which Terence Tao called “perhaps the most unambiguous instance” of AI solving an open problem. Harmonic, the math AI startup backed by Robinhood CEO Vlad Tenev, claimed to crack an Erdős problem open for 30 years. Even programming legend Donald Knuth was left stunned when Claude solved an open problem he had been working on for weeks.
But the Sawin result points to something the AI-in-math narrative has underemphasized: breakthroughs by AI models don’t just advance mathematics directly — they expose new territory for human mathematicians to explore. Once the OpenAI model showed that polynomial improvement was possible, and outlined a general strategy for achieving it, the subsequent question — “how good can the exponent get?” — became a well-posed problem that a skilled human could attack directly. Sawin attacked it and won.
This is not AI versus humans. It looks more like AI clearing a conceptual bottleneck and humans sprinting through the opening.
The Remaining Gap And What It Means
Sawin’s paper also addresses the theoretical ceiling of the approach. Using a separate argument, he shows that the methods used — constructing lattices from CM number fields and counting short projections — cannot, even in principle, exceed an exponent of about 1.243. The true upper bound on the unit distance problem remains n^(4/3), due to Spencer, Szemerédi, and Trotter. The gap between the best known lower bound (now 1.014) and the best known upper bound (1.333) remains large, but it is no longer infinite in the sense that mattered most: for 80 years, no one had shown the exponent could exceed 1 at all.
The fact that a human mathematician, working without AI assistance, was able to improve on an AI-generated breakthrough within days — and improve on it dramatically — says something about how this new era of mathematical research may actually function. AI models identify surprising connections. Human mathematicians, now alerted to the right structure, figure out how to make those connections sing.
Mathematics is increasingly a collaboration between pattern-finding machines and the humans who understand what the patterns mean. On the Erdős unit distance problem, that collaboration is already yielding results faster than anyone expected.