Place a handful of dots on a sheet of paper and measure the distances between them. Rearrange the dots — how many pairs can you get sitting exactly the same distance apart? Simple enough to sketch on a napkin, and for 80 years, mathematicians believed they knew the best possible answer, thanks largely to Paul Erdős, one of the 20th century’s most prolific mathematical minds.
Then an AI model came along and quietly did better.
A deceptively simple puzzle with an 80-year history
The planar unit distance problem sounds almost trivial. Place dots on a flat plane, measure the distances between every pair, and find the arrangement that maximizes how many pairs share exactly the same distance. The challenge is not in understanding the question — it is in finding the best possible answer.
Paul Erdős first posed the problem in 1946. His argument was that a grid-like arrangement of dots was optimal, and that the number of equal-distance pairs could only be slightly larger than the total number of dots. That conjecture shaped how mathematicians thought about the problem for generations.
For eight decades, very little progress was made. No one proved Erdős right, but no one proved him wrong either. By 1982, Erdős put money on it — $300 for a proof or disproof, later raised to $500 around 1995. Cash prizes, for Erdős, were a signal of genuine mathematical seriousness.
The eccentric genius behind the problem
Paul Erdős was born in Budapest in 1913 to two mathematics teachers. His childhood was marked by loss: both older sisters died of scarlet fever on the day of his birth, and his father spent six years in a Siberian prisoner-of-war camp after World War I began. Numbers, Erdős later recalled, became his friends — reliable and unchanging in a world that was neither.
After earning his PhD, Erdős abandoned any fixed address and traveled constantly, staying with fellow mathematicians around the world, often arriving unannounced with: “My brain is open!” His output was staggering — more than 1,500 co-authored papers. His reach was so wide that other mathematicians began tracking their “Erdős number,” counting how many collaborative steps separated them from him. Albert Einstein holds an Erdős number of two.
Erdős also attached cash prizes to hundreds of unsolved problems, ranging from $10 to $10,000. The system was famously informal — he sometimes forgot which prizes he had offered.
How an AI model found what humans missed
The announcement came with little fanfare. OpenAI revealed that one of its internal models had identified a new strategy — rooted in algebraic number theory — for arranging dots on a plane that produces more equal-distance pairs than any arrangement Erdős had proposed.
Why did an AI succeed where humans had not? The companion paper offers a candid explanation: human mathematicians tend to work within narrow specialties, while the model could draw on knowledge spanning multiple domains at once. Most researchers had also simply assumed Erdős was right, so few devoted serious effort to disproving him.
The result did not stop there. Mathematician Will Sawin, at Princeton University, built on the model’s approach to find an even better solution. Sawin, Thomas Bloom, and colleagues then used related techniques to crack another long-standing open problem — the sum-product conjecture, which Erdős had posed in the 1970s. Sawin’s framing was measured: “It’s not that AI solved an impossible math problem, but it’s not nothing. It’s somewhere in between.”
Transparency gaps and a cautious math community
Not everyone received the announcement with uncomplicated enthusiasm. OpenAI did not release the model’s original output — the company published a rewritten summary of the model’s reasoning alongside a proof revised by human mathematicians. What the public saw was not the AI’s raw work, but a polished reconstruction of it.
This was not the first time OpenAI had made bold claims about Erdős problems. A previous announcement declared that GPT-5 had found solutions to ten unsolved Erdős problems. In reality, the model had surfaced existing solutions already buried in the literature. Thomas Bloom, who maintains the Erdős Problems website, called it a “dramatic misrepresentation.”
This time, Bloom was part of the companion paper. His conclusion was careful: “The human still plays a vital role in discussing, digesting and improving this proof, and exploring its consequences.” Rodrigo Ochigame, an anthropologist at Leiden University, noted that OpenAI disclosed nothing about the methods used, prompts given, training data, or computational resources consumed — making independent scientific assessment essentially impossible.
The Leiden Declaration and the ethics of AI in mathematics
On June 2, a group of mathematicians published the Leiden Declaration on Artificial Intelligence and Mathematics, laying out ethical guidelines for researchers, policymakers, and organizations navigating AI’s growing role in the field.
The declaration asks researchers to disclose which AI tools they used, take responsibility for accuracy, and properly cite existing scholarship an AI may have drawn upon. That these baseline standards needed stating explicitly reflects how quickly the landscape has shifted.
The authors also raise a harder question about motivation. AI companies pursue mathematical problems to improve the reasoning capabilities of commercial models — not out of love for the discipline. Some models are trained on mathematicians’ own published work, then deployed for applications raising serious ethical concerns, including warfare and mass surveillance, with the people whose intellectual labor trained these systems often having no say in how the resulting tools are used.
Erdős spent his life believing mathematical truth belonged to everyone willing to search for it. Whether that spirit survives the age of proprietary AI models is a question worth sitting with.