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Sat, 30 May 2026 16:58:23 -0700
marlon from private IP, post #12549159
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A Famous Math Problem Stumped Humans for 80 Years. AI Just Cracked It.
https://www.wsj.com/tech/ai/ai-math-solves-erdos-problem-openai-c4029e84
A Famous Math Problem Stumped Humans for 80 Years. AI Just Cracked It.
The math world is losing its mind over the new solution to an Erdős problem. This is what AI found, how we missed it—and why it matters.
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Emil Lendof/WSJ
Ben Cohen
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Ben Cohen
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May 29, 2026 9:00 pm ET
“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 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 this prompt:
It spit out this proof:
And everyone in math lost their minds.
For those who aren’t fluent in numbers, 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 elementary 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:
“AI was able to do here what lots of excellent human researchers tried and failed to do.”
— Noga Alon, professor, Princeton University
“This is the first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator.”
— Daniel Litt, assistant professor, University of Toronto
“There is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics: 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.”
— Timothy Gowers, professor, Collège de France
The last endorsement was especially weighty coming from a winner of the Fields Medal, one of the highest honors for human mathematicians. Even if AI never gets
smarter, Gowers continued, we have already entered a new era.
“It will become very difficult for humans to compete with AI at solving mathematical problems,” he said.
Just looking at formulas is enough to hurt my brain, but 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.
When I spoke with OpenAI employees, they 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 posed by Paul Erdős, known as the most prolific mathematician in history.
He was also known as an eccentric, nomadic genius who lived out of a suitcase, worked around the clock and traveled all over the world, true to his personal
motto: “Another roof, another proof.”
In addition to his research, he left behind a sprawling collection of questions known as Erdős problems, which have become a benchmark for measuring progress
in math.
You can tell how much he liked any given problem by the amount of money he offered for its solution. The unit distance problem was among his favorites: It
originally came with a $300 bounty, which Erdős later bumped up to $500.
When he wasn’t assigning monetary values to math problems, Erdős divided them in two categories: marshmallows (“a tasty tidbit supplying a few moments of
fleeting enjoyment”) and acorns (“requiring deep and subtle new insights from which a mighty oak can develop”).
This problem was a giant acorn—and OpenAI wanted to crack it.
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?
Erdős 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 OpenAI research team, from left to right: Sebastien Bubeck, Mehtaab Sawhney, Mark Sellke, Hongxun Wu, Alex Wei and Lijie Chen.
The OpenAI research team, from left to right: Sebastien Bubeck, Mehtaab Sawhney, Mark Sellke, Hongxun Wu, Alex Wei and Lijie Chen. OpenAI
OpenAI’s researchers were stunned. They had given this Erdős problem to an internal model as a test of its capabilities—to find out whether it was better
than previous models. They found out how much better it was once they took a peek at the solution. “I initially didn’t believe it,” said Mehtaab Sawhney,
a Columbia mathematician at OpenAI. So they searched for errors, verified the results with outsiders and checked the AI’s work using the company’s AI coding
agent. “With enough reading and enough Codexing,” Sawhney said, “it seemed believable—and pretty remarkable.”
Long before AI, mathematicians who solved Erdős problems often framed their checks instead of cashing them. For them, the money was worth less than the glory.
When I asked OpenAI researchers about their plans for the prize, they hadn’t given it much thought.
But they did have lots of thoughts about my next question: 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 Erdős’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.
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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 Erdős 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
at OpenAI. “So you give up and move on.”
AI doesn’t move on. It keeps plugging away without taking 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—the length of the
first “Harry Potter” book.
The unit distance problem: n dots, exactly one unit apart.
The unit distance problem: n dots, exactly one unit apart. OpenAI
After reading it, a former OpenAI researcher did some back-of-the-envelope math and estimated 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.”
You might find all this craziness upsetting, or inspiring, or both, but the people inside OpenAI are surprisingly optimistic about the future of
mathematicians.
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 and using them to take
down other longstanding mathematical problems.
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“The point of a breakthrough,” Bubeck wrote on X, “is that suddenly it makes a lot of things that seemed impossible possible.”
Of course, the ability to solve Erdős problems isn’t the same as superhuman intelligence, wonky math research isn’t a cure for cancer and the miraculous
era of AI-powered discovery isn’t quite here.
“It’s fair to say that we haven’t seen yet the spark of genius that you could attribute to some of the grandest proofs in the history of humanity,”
Bubeck told me.
But maybe we should get used to sitting down. It’s also fair to say that AI is perfectly capable of powering scientific progress in any field with problems
waiting to be solved.
And now there is proof. Or disproof.
Tue, 02 Jun 2026 14:27:52 -0700
doublefriedchicken from private IP
Reply #16066727
This was proved with a counterexample. I think another proof is that every even number is the sum of at two primes. It's never been proven and there are
endless numbers to check. For example, 30 is the sum of 17 and 13.
Wed, 03 Jun 2026 07:44:20 -0700
phosita from private IP
Reply #10036719
If someone/something figured out prime factorization then the fecal material would hit the air circulation apparatus.
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