An 80-Year-Old Math Conjecture, Overturned by GPT
An 80-year-old open problem in mathematics — the planar unit distance conjecture — has just been overturned by GPT-5. Fields Medalist Timothy Gowers described his reaction with deeply mixed feelings.
In a tweet, Gowers wrote: "If you are a mathematician, you might want to make sure you're sitting down before you read on." The moment was widely compared to witnessing a seismic shift in the field.
In the same period, GPT also helped resolve another long-standing open problem: the point convergence of Nesterov's Accelerated Gradient (NAG) method — a question that had remained unanswered for 40 years.
It seems AI's mathematical capabilities no longer need benchmark exams to prove themselves.
The Planar Unit Distance Conjecture
The problem, posed by Erdős in 1946, is deceptively simple: Scatter n points on a plane. What is the maximum number of pairs of points whose distance is exactly 1?
Erdős conjectured that the square lattice arrangement is already optimal — no configuration could yield a substantially higher count. Because the conjecture aligns so naturally with intuition, it became known as "the most famous and easiest-to-explain open problem" in combinatorial geometry. Yet for 80 years, no one could prove or disprove it.
GPT found an entirely new family of point arrangements in which the number of unit-distance pairs exceeds the lattice scheme by a polynomial factor — definitively resolving the conjecture. The proof's path was unexpected: the model constructed an infinite tower of totally real number fields and completed the argument using Galois groups and the Golod–Shafarevich theorem.
Fully Automated Solution
Gowers stated he would "unhesitatingly recommend it for publication." Nine mathematicians — including Noga Alon, Thomas Bloom, and Melanie Wood — co-authored a companion paper to independently verify the proof.
The NAG Point Convergence Problem
The second breakthrough concerns Nesterov's Accelerated Gradient (NAG) method, invented in 1983. NAG improves standard gradient descent by "peeking" one step ahead before deciding how to move, boosting the convergence rate from O(1/k) to O(1/k²) — a full order of magnitude faster.
But one question remained open for 40 years: Do the iterates of NAG actually converge to a specific point, or do they orbit the optimum indefinitely?
Ernest Ryu of UCLA and his student Uijeong Jang used GPT-5 Pro to prove the answer: yes, NAG converges — and to a definite point.
Point Convergence of NAG — paper abstract mentioning ChatGPT
Ryu also shared the full ChatGPT conversation used during the proof process:
The shared ChatGPT conversation log
https://chatgpt.com/share/6950b63e-1a58-8009-832b-48288fd60c30
In a follow-up, Ryu used an internal OpenAI model to prove a related result: the continuous-time NAG (Nesterov flow) can take an infinitely long path to converge to a minimizer. In the paper's acknowledgments, he wrote: "All proofs of this work are due entirely to an internal model at OpenAI."
Follow-up paper — all proofs completed entirely by OpenAI's internal model
Paper: Point Convergence of NAG
https://arxiv.org/abs/2510.23513
Follow-up Paper: Nesterov Flow May Travel Infinitely Long
https://arxiv.org/abs/2604.06651
https://chatgpt.com/share/6950b63e-1a58-8009-832b-48288fd60c30