A Math Problem Stumped Everyone For 80 Years. AI Just Cracked It. --- The math world is losing its mind over a proof for one of the Erdos problems

“"If you are a mathematician," one of the world's leading mathematicians recently wrote, "you may want to make sure you are sitting down before reading further."

 

And you'll definitely need to sit down if you're not a mathematician.

 

Because a famous math problem that stumped humans for the better part of a century has finally been toppled -- by AI.

 

Not long ago, the most advanced AI models couldn't do basic math. By last year, they were performing at gold-medal levels at the International Mathematical Olympiad. Now, they are solving classic problems in combinatorial geometry using algebraic number theory. In no time at all, artificial intelligence has gone from stupid to frighteningly smart.

 

But even mathematicians were astonished when OpenAI recently announced that one of its models resolved a puzzle known as the unit distance problem without the help of any humans scribbling a bunch of equations on chalkboards.

 

It was fed a prompt and it spit out a proof. And everyone in math lost their minds.

 

OpenAI helped translate its findings by presenting them alongside 19 pages of companion remarks from prominent mathematicians.

 

As a rule, mathematicians are severely allergic to hype. They demand proof before they are willing to accept basic facts, much less claims about novel breakthroughs, and many of them have been skeptical about AI revolutionizing their industry.

 

So it was startling to read stuff like this: "If a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that."

 

That endorsement was especially weighty coming from Timothy Gowers, professor, College de France, a winner of the Fields Medal, the highest honor for human mathematicians. Even if AI never gets any smarter, Gowers continued, we have already crossed a line.

 

"It will become very difficult for humans to compete with AI at solving mathematical problems," he said.

 

I wanted to know more about what the AI found, how we humans missed it -- and why this breakthrough matters to those of us who would like to permanently distance ourselves from math problems.

 

OpenAI employees told me this result would have sounded completely bananas one year ago.

 

"Forget one year ago," researcher Sebastien Bubeck said. "A month ago."

 

So imagine how unimaginable it was 80 years ago, when the unit distance problem was first posed by Paul Erdos, known as the most prolific mathematician in history.

 

He left behind a sprawling collection of questions known as Erdos problems, which have become a benchmark for measuring progress in math. The unit distance problem was among his favorites.

 

The simplest version of the unit distance problem goes something like this: If you put n dots on a sheet of paper, how many pairs of dots can be exactly one unit apart?

 

Erdos showed in 1946 that arranging those dots in a grid produced a certain number of pairs, and his conjecture was that no arrangement could do much better. OpenAI's model found one that does. In other words, the proof was a disproof. The construction it discovered is abstruse, but here's the upshot: It yields more pairs than Erdos or anyone else envisioned.

 

Including the team at OpenAI.

 

When the researchers pointed a general-purpose reasoning model at the trickiest Erdos problems to test its capabilities, it soon became apparent that the internal model was far more capable than they realized.

 

First, the model's solution went to an AI grading system, which believed it was correct. Only then did the humans take a peek. "I initially didn't believe it," said Mehtaab Sawhney, a Columbia mathematician currently at OpenAI. They showed the results to external mathematicians for proper verification. They also checked the AI's work using the company's AI coding agent.

 

I asked the researchers: Why did AI succeed where humans failed?

 

The first explanation is that this particular solution happens to be highly counterintuitive.

 

Most people who tackled this problem tried to prove Erdos's conjecture, rather than disprove it. Only by defying conventional wisdom and experimenting with seemingly improbable strategies did the model find an unexpected path forward.

 

The second is that humans specialize while AI synthesizes.

 

While mathematicians tend to focus on their specific areas of expertise, AI models use their vast knowledge to spot connections that we couldn't possibly see ourselves. In this case, that meant pulling from both algebraic number theory and discrete geometry, which have about as much in common as the marathon and pole vault.

 

The third explanation is that AI has time, attention, patience, focus and the persistence to stick with methods that humans might abandon -- and the solution to this Erdos problem demanded it.

 

"It's the kind of idea that you try for a bit, it doesn't work, and you think maybe you were just too hopeful," said Mark Sellke, a Harvard statistician on leave at Open-AI. "So you give up and move on."

 

AI doesn't move on. It keeps plugging away without taking any breaks to eat, sleep, answer emails, pick the kids up from school and watch the Knicks.

 

And it can think coherently for so long that even an abridged version of the model's "chain of thought" ran more than 75,000 words.

 

After reading it, a former Open-AI researcher did some back-of-the-envelope calculations and estimated that it took less than 32 hours and $1,000 in tokens, a bargain for a result of this caliber.

 

The researchers wouldn't confirm the exact amount of time and compute, but Bubeck described the costs as "really nothing crazy at all."

 

Whether you find all of this craziness upsetting or inspiring -- or both -- depends on how you feel about AI. The people inside OpenAI are surprisingly optimistic about the future of the mathematicians who just had their minds blown.

 

They point to domains where unthinkable technological advances have improved human performance, from Go players to chess grandmasters. Like a calculator, they say, AI is a tool that can expand our curiosity rather than destroy it. In fact, humans are already building on this solution's methods, refining them, strengthening them and using them to take down other longstanding mathematical problems.

 

"The point of a breakthrough," Bubeck wrote on X, "is that suddenly it makes a lot of things that seemed impossible possible."

 

It's also fair to say that AI can power real scientific progress in any field with problems waiting to be solved.

 

And now there is proof. Or disproof.” [1]

 

1. EXCHANGE --- Science of Success: A Math Problem Stumped Everyone For 80 Years. AI Just Cracked It. --- The math world is losing its mind over a proof for one of the Erdos problems. Cohen, Ben.  Wall Street Journal, Eastern edition; New York, N.Y.. 30 May 2026: B2.