An 80-year-old mathematical dead end? Solved.
Not by a genius in a tweed jacket. Not by a decades-long collaboration of Ivy League professors. By a straightforward query to a chat bot.
OpenAI announced it yesterday. The math world is reeling. Experts called the method “clever” and “elegant.” Previous AI feats in math? Mostly noise. This is signal. This is a proof good enough for the top journals, even if a human wrote it.
“No previous AI-generated proof has come close,” Timothy Gowers wrote. He’s at Cambridge. He means it.
Daniel Litt, from Toronto, was brought in to verify it. He agrees. “This is the unique interesting result produced autonomosuly by AI so far.”
The problem is simple. Almost childish.
Draw dots on paper. Try to get as many pairs as possible to be exactly one inch apart. Nine dots? Easy. Put them in a grid. You get 12 pairs. But what if you have billions of dots? Trillions?
In 1946, Paul Erdős guessed the best way. He thought the answer lay in a tight, carefully spaced grid. He proved you could get slightly more than a standard grid. He also claimed that’s the limit. The ceiling.
Nobody beat him for eight decades.
Nobody proved he was right, either.
Most mathematicians believed him. They tried to prove his conjecture. They hit walls. Two weeks ago, OpenAI fed the problem to an internal language model. They asked, essentially: Is Erdős right?
The AI churned. Hundreds of pages of logic. And then, it broke the record.
It didn’t follow Erdős’s grid.
“It feels like magic,” said researcher Sawhney.
The AI built a shape. Not a flat grid. A higher-dimensional lattice. Weird geometry with special symmetries. Then, it smashed that high-dimensional structure flat onto the page. A numerical shadow. You couldn’t draw it, not really. Too complex. Too tangled.
Did the AI find the ultimate solution? Probably not.
Will Sawin, a mathematician, already improved on the AI’s work. Just by tweaking it.
But here’s the rub. OpenAI contacted top mathematicians—Gowers, Litt, Bloom—to check the proof. They didn’t see the AI’s raw output. They saw a cleaned-up version. They agreed. The logic held.
Why did it work?
Patience.
Humans give up. We see a dead end. We turn away. An AI doesn’t get frustrated. It just tries. And tries. And tries in “treacherous waters” without flinching.
“They can play for longer,” says Jacob Tsimerman. “Without getting overwhelmed.”
Most mathematicians thought Erdős was right. So they tried to prove him. The AI looked for a counter-example. It found one.
Was this luck?
Maybe.
Daniel Litt suggests the AI got lucky. It stumbled upon a case where experts had looked, blinked, and missed a simple approach. The tools existed. Humans just didn’t use them in this weird, high-dimensional way.
Groundbreaking ideas? Still human territory. But “rare gems”? Those are popping up.
There’s a darker side, though.
The AI doesn’t cite sources. It presents borrowed ideas as its own. Melanie Matchett Wood, at Harvard, warns this is dangerous. For a human, that’s plagiarism. For an AI? Just standard operating procedure.
“We recognize very similar ideas in the literature,” Wood says. They weren’t credited.
Should we fix this? The community has to decide. Fast. The world changed since December.
“Any mathematician who hasn’t been using the latest models should be surpirised.”
Maybe people spent too long being polite to Erdős’s legacy. Maybe we needed to play devil’s advocate. The AI did it. It found the hole in the theory we couldn’t see because we believed in the wall.
Are these moments common? We’re about to find out.
The door is open. No one knows what’s on the other side yet.
