待翻譯：Why the Legendary Erdős Problems Are Falling to AI

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Read Later Copied! Comments Read Later artificial intelligence Why the Legendary Erdős Problems Are Falling to AI August 3, 2026 AI’s greatest mathematical successes have come from answers to problems posed by a mid-20th century iconoclast. By examining what makes the Erdős problems unique, mathematicians are trying to understand how AI might change the rest of math. Read Later DVDP for Quanta Magazine On May 20, 2026, OpenAI made an announcement that shook the mathematical world. An internal AI model — one not available to the public — had come up with a counterexample to the “unit distance” problem, a conjecture made in 1946 by Paul Erdős, the prolific, itinerant Hungarian mathematician. Erdős posed thousands of questions, but this one was special: It was both simple to explain and mathematically deep. It was the first historically significant proof to come from an AI model. Though the model’s result wasn’t definitive — human mathematicians would substantially improve on it within weeks — it was innovative, bringing in ideas from a distant branch of math that no one had successfully applied to this problem before. And it was influential: Within a few days, related techniques were used to solve other important problems. Then on August 1, OpenAI announced that an unreleased model named Astra made 10 additional mathematical advances, including finding solutions to three more problems posed by Erdős. Many mathematicians have hailed developments such as these as a phase transition in the mathematical capability of AI models. These models are “changing dramatically the way mathematical research is being done,” said Noga Alon of Princeton University, who has solved dozens of Erdős problems over his decades-long career. Erdős and his conjectures have long fascinated mathematicians. He traveled constantly — living out of a suitcase for years at a time, staying with friends, owning almost nothing. He rattled off problems in published papers and letters to mathematicians around the world, often attaching prize money that he would pay out of pocket to the first person to come up with a solution. The reward might be a token $10 or $25, or, for problems he considered important or difficult, it could range into the thousands. Erdős died of a heart attack in 1996 while attending a math conference in Warsaw, but a nonprofit foundation based in Iowa has promised to make good on his bounties. He was a beloved figure, but also a downright weird one. He only wore silk, and he avoided the touch of other people. Deeply cynical about authority, he gave away most of the money he earned and relied on a friend to manage his finances and other practical affairs. He referred to God as the “Supreme Fascist” and fueled his incessant output of mathematical ideas with a steady diet of amphetamines. It is a strange irony of history that the problems he suggested have now become a central proving ground — and, in effect, a series of PR coups — for the world’s biggest and most powerful technology companies. But in all likelihood none of this would have happened had it not been for an English mathematician named Thomas Bloom. Many Meetings Like Erdős, Bloom was interested in both number theory and combinatorics. His focus has been an area called arithmetic combinatorics, which lies at the intersection of the two. After getting his doctorate in 2014, Bloom established himself as a rising star in the field, landing a prestigious fellowship from Britain’s Royal Society, which let him work at almost any university he wanted to. (He’s now at the University of Manchester.) Bloom has liked Erdős’ style for as long as he can remember. But he always found it hard to keep track of which problems had been solved and which had been forgotten entirely. So in early 2023, he decided to gather as many problems as he could into a list. He intended it for his own use. But “I thought it would be easier if I could access it wherever I was,” he said; he figured he “might as well make a website, kind of with the expectation that maybe nobody would use it.” He gathered a couple hundred problems and launched erdosproblems.com. Bloom used ChatGPT to write the Python code that ran the website, which was, at the time, a remarkable thing for a large language model to be able to do. Using one to collaborate on the math itself still seemed like only a distant possibility. His goal was not just to cross items off a list. He wondered if “modern day mathematics, often using techniques unknown by Erdős, could clear up many of these more obscure problems,” he wrote in a blog post. “We will then be left with a core of interesting, difficult problems, which can serve to demonstrate the limits of our knowledge.” Bloom did crucial work in curating the list: Sometimes Erdős stated problems in ambiguous or unclear ways, and Bloom figured out what the most sensible version of each problem should be. He kept adding problems to the site, and gradually its audience grew. Over the course of 2024 and the first eight months of 2025, the statuses of 111 problems on the list were changed from “open” to “solved” (although some of these had been solved years earlier, and their status change reflected the rediscovery or verification of a proof). Then, in August 2025, some colleagues suggested that Bloom add a commenting function, so that people could talk about problems they were interested in. He was able to do so quickly, using ChatGPT to write the code. By now he’d cataloged nearly 1,000 problems. Bloom’s timing was good. He made it possible for like-minded people to talk to one another, and that “really let a community build up,” he said. For the most part, comments were sporadic — a problem might attract a single comment pointing out an example or noting how hard the problem looked. But activity steadily grew, and some problems catalyzed nuanced mathematical discussions between strangers. “Tom probably never really realized this, but for me it’s honestly changed my life,” said Wouter van Doorn, the fourth-most-prolific commenter on Bloom’s website. Like many people who became active on the site in the autumn of 2025, van Doorn isn’t exactly a professional mathematician. He works “for a company that gets hired by other companies to do customer service support,” as he put it. But he isn’t exactly an amateur either — a decade prior, he almost completed a master’s degree in math at KU Leuven in Belgium. In 2024, spurred in part by how capable he saw LLMs getting, he took a six-month leave of absence from work to focus on math. At the time, while he didn’t particularly want to use AI, he remembers thinking, “Right now I’m still better at mathematics than an AI is, but who knows what it’ll be in a year, two years, five years? If I want to finish these projects, and I want them to be mine, now is the time.” And so, in October 2025, van Doorn, now back at his day job, left the first comment on the page for Problem 1102. The problem, which Erdős posed in 1981, asks about properties of sets of “square-free” integers — that is, integers that have no repeated prime factors. (For instance, 30 is square-free because it is equal to 2 × 3 × 5, but 18 is not, because it is equal to 2 × 3 × 3; the 3 repeats.) In early November, van Doorn shared progress toward an answer — which he’d figured out without relying on AI — as a comment on the problem page. Later that day, another commenter on the site replied, claiming he had found a flaw in van Doorn’s argument. The two traded remarks in rapid succession, and van Doorn convinced his interlocutor that his argument was correct. “I see how your argument works now. Nice!” the other mathematician replied. That other mathematician was Terence Tao, a professor at the University of California, Los Angeles who is arguably the best-known mathematician alive today, and inarguably one of the most influential. (Not incidentally, when Tao was just 10 years old, he crossed paths with Erdős.) Bloom’s website, which has the look and feel of an earlier time, was becoming an example of the internet at its democratic best. “This entire collaboration would not have been possible without Tom’s website and the comments section there,” van Doorn said. It didn’t matter if you had tenure or not, if you were young or old, if you were at a fancy university or even at a university at all. If you wanted to work on math and had good ideas, you could find people to collaborate with. But as the winter set in — around the same time that van Doorn found himself collaborating with Terry Tao — things started to change. Journey to the Cross-Roads Kevin Barreto and Liam Price, both in their early 20s, became friends in the summer of 2025 on a Discord server dedicated to AI. Barreto is currently an undergraduate at the University of Cambridge; Price studied some math in college but left before finishing. In December, convinced that the newest AI models might succeed in resolving some Erdős problems, the pair started throwing batches of problems at them. They realized early on that if they told GPT-5.2 that a problem’s answer wasn’t known, it wouldn’t make much headway, so as Barreto put it, they learned how to “prompt it in a very particular way, gaslighting it into thinking the problem is easier than it actually is.” They had what they thought was their first triumph on Erdős Problem 333. Early on Christmas morning, Barreto posted a proof to Bloom’s website, writing, “We believe, to the best of our knowledge, this is the first case of an LLM fully autonomously resolving an Erdős problem, not previously resolved by humans.” Even though 333, which dealt with the sums of sets of integers, was not a particularly important problem, solving it with AI still felt important. But a few hours later, another user pointed out that Erdős himself had provided a resolution to 333 in a paper published in 1977. Barreto owned up to the mistake. “My formal request to all members of the website is to put greater focus on literature search on the problems currently marked as open,” he wrote. “As someone who has fallen for this twice now, it’s quite gut-wrenching.” Undeterred, he and Price kept at it, and by January 4, 2026, they’d used GPT-5.2 Pro to find a solution to Erdős 728, a problem about when certain numbers are divisible by other numbers. This time nobody could find prior work already proving it. Barreto used another AI tool called Aristotle (developed by a startup called Harmonic) to certify that the proof held together logically. Nat Sothanaphan, a software engineer and the only forum participant more prolific than Bloom, Tao, and van Doorn, had ChatGPT write up the formalized result and posted it online. Price developed a methodology for how to ask LLMs to solve open questions. First, he would ask a chatbot for a solution. Then he would feed that solution into a fresh instance of the chatbot, asking it to check the previous chatbot’s work. He’d repeat this process until he had what looked like a workable solution. (This echoes some of the work that companies have been doing internally to create what they call harnesses or scaffolds, which automate the sort of iteration that Price does by hand.) Barreto and Price’s papers represent just a fraction of the many Erdős problems solved at least in part by AI over the past few months. There are multiple reasons why these problems in particular have become such a fertile test bed for LLMs. The primary one is that, by and large, Erdős problems are in number theory, combinatorics, and graph theory, all areas of math that have proved more accessible than others to large language models. The problems also vary widely in difficulty and mathematical significance. This variation makes them appropriate for a nascent technology whose abilities also vary widely. Many of Erdős’ problems had a monetary value attached to them from their moment of inception, a playfu [truncated for AI cost control]