ChatGPT solves – OpenAI says its large language model settled an 80-year-old geometry conjecture—showing a new way to place points so more pairs end up exactly one inch apart. Mathematicians called the method “clever” and “elegant,” while emphasizing that humans still had to v
When a problem can survive 80 years of human effort, even a hint of progress feels electric. Yesterday. OpenAI announced that a chatbot helped resolve the “unit distance” version of an Erdős conjecture—an achievement that mathematicians say would normally earn a place in a top math journal and major media attention. even if the proof had been produced by people instead of an AI.
The claim arrived with an unusual mix of astonishment and scrutiny. Several experts consulted by OpenAI praised the artificial intelligence’s approach as “clever” and “elegant.” Timothy Gowers. a mathematician at the University of Cambridge. wrote in commentary solicited by OpenAI that no previous AI-generated proof has come close to meeting those high standards.
The geometry challenge itself is deceptively simple to describe. Draw nine dots on a sheet of paper. The goal is to get as many pairs of dots as possible to be exactly an inch apart. You can put all the dots in a line and get eight such pairs. Or you can arrange them as a three-by-three grid, yielding 12 pairs. For any number of dots—whether you’re thinking of hundreds. billions. or trillions—the question is the same: what’s the highest number of pairs you can achieve at distance one?.
In 1946, mathematician Paul Erdős made a guess about the best strategy. His approach used a grid, but with much smaller spacing between dots, allowing pairs to form across several grid points. Erdős also proved that with extremely careful choice of spacing—backed by sophisticated mathematics—you could do slightly better than a simple grid. Then he claimed that no one could do better than his construction.
Erdős’s conjecture resisted proof for decades. Over eight decades, human mathematicians made valiant efforts but neither proved Erdős right nor managed to disprove him. Many experts agreed with Erdős’s intuition, but certainty never arrived.
That changed two weeks ago, according to OpenAI’s account. OpenAI mathematicians Mehtaab Sawhney and Mark Sellke fed the conjecture into an internal large language model trained for general reasoning. They asked the system whether Erdős was right. The model generated hundreds of pages of careful logic and calculations—and. in doing so. it surpassed Erdős’s long-standing record.
For the people who watched it happen, the experience sounded almost surreal. “It feels like magic,” Sawhney said. “It’s kind of an amazing experience to have a machine give back something which really resembles how I work.”
Sellke described how the AI’s solution differed from the familiar “square grid” approach. “What the model did is totally different from the ‘square grid’ construction.” Instead. it built a more elaborate grid that lives in a kind of higher dimension. That higher-dimensional lattice of points had special mathematical symmetries, which helped separate even more pairs at the same distance.
The model then developed a way to map that higher-dimensional structure back down to the two-dimensional page. producing what Sawhney described as a flattened numerical “shadow.” The result. he said. isn’t a grid you can straightforwardly draw on paper; even for a small number of dots. it would be “too difficult to actually draw on paper.”.
Still, the AI did not prove that its approach is the absolute best possible. In fact, Will Sawin—already a mathematician active in this area—has improved upon the AI’s grid.
OpenAI didn’t keep the work sealed away. The company privately contacted Daniel Litt, Will Sawin, Timothy Gowers, and a number of other mathematicians to verify the proof. Those experts, working together and without the company’s direct involvement, wrote up their individual takeaways. Importantly, no external experts have seen the AI’s original output—only an edited version of its train of thought.
What stood out to the mathematicians who evaluated it was not just the outcome. but the discipline of how it got there. They said the AI showed “preternatural patience and focus.” Human experts had often spent effort trying to prove Erdős rather than trying to break the conjecture. and even those few who looked for a counterexample would likely have found the route too difficult and tedious—especially the construction of the high-dimensional shape—without any particularly encouraging sign that it would work.
A different kind of machine advantage was part of the explanation offered by others in the field. Jacob Tsimerman. a mathematician at the University of Toronto who was not involved in the work but contributed to a companion paper solicited by OpenAI. argued that AIs can try longer and in more treacherous terrain than mathematicians “without getting overwhelmed.”.
Experts also pointed to a key feature of unit-distance problems: while the problem is well known. a proof confirming Erdős would likely contain mathematically rich new ideas that could transfer to other questions. They said the AI’s method doesn’t appear to introduce fundamentally brand-new tools no one could have anticipated. “The model did not invent something fundamentally new that nobody saw coming. ” said Sébastien Bubeck. a mathematician leading OpenAI’s mathematical explorations. “It just executed like an amazing mathematician.”.
And even when the machine’s work lands, the human standard still matters. Thomas Bloom. in a “reflections” document. wrote that without human intervention to “clean up” the AI’s work. the result wouldn’t be convincing. The proof still had to be discussed, digested, and improved—and its consequences explored.
Not everyone frames the breakthrough as a triumph of creativity. Melanie Matchett Wood. a Harvard University mathematician. suggested that progress may have been limited by experts’ belief that the conjecture was true. If the community that reviewed the LLM’s answer had instead spent the same amount of time seeking a counterexample. she said. they would likely have found one. “Maybe people should be spending more time, you know, playing devil’s advocate,” Wood said. She also provided a commentary for OpenAI.
One plausible reason, Daniel Litt said, is that the AI may have located a case experts tried and missed. “I guess it got lucky that it found one of the cases where experts tried and missed something,” Litt said. And because genuinely new. groundbreaking ideas remain beyond the reach of current LLMs. he suggested. the machines may be most effective at mining the literature for rare gems—approaches humans didn’t attempt or didn’t follow through on.
Yet Wood added a warning that sits uncomfortably beside the celebration: mathematicians now have to contend with the way AI systems sometimes treat academic credit. She said the team recognized “very similar ideas in the literature that weren’t credited.” In her view. if a human had reused those ideas without credit. it would be “professional malpractice.” Wood believes the community needs to decide quickly how to handle AI’s nonadherence to academic norms. because “things are changing fast.”.
Wood also warned that the pace of change is real enough to catch people off guard. “Any mathematician who hasn’t been using the latest models should be surprised,” she said. “It’s quite a different world than in December of last year.”
The net result is a milestone in geometry—and a jolt to the culture around mathematical proof. The AI found an answer that humans failed to lock down for 80 years. But what convinced the field wasn’t just the computation. It was the way experts were able to verify. scrutinize. and situate the work within the standards that govern what counts as mathematics.
OpenAI ChatGPT unit distance problem Erdős conjecture geometry large language model mathematics Timothy Gowers Daniel Litt Mehtaab Sawhney Mark Sellke