Amateur armed with ChatGPT solves an Erdős problem

Liam Price just cracked a 60-year-old problem that world-class mathematicians have tried and failed to solve. He’s 23 years old and has no advanced mathematics training. What he does have is a ChatGPT Pro subscription, which gives him access to the latest large language models from OpenAI.

Artificial intelligence has recently made headlines for solving a number of “Erdős problems,” conjectures left behind by the prolific mathematician Paul Erdős. But experts have warned that these problems are an imperfect benchmark of artificial intelligence’s mathematical prowess. They range dramatically in both significance and difficulty, and many AI solutions have turned out to be less original than they appeared.

The new solution—which Price got in response to a single prompt to GPT-5.4 Pro and posted on www.erdosproblems.com, a website devoted to the Erdős problems, just over a week ago—is different. The problem it solves has eluded some prominent minds, bestowing it some esteem. And more importantly, the AI seems to have used a totally new method for problems of this kind. It’s too soon to say with certainty, but this LLM-conceived connection may be useful for broader applications—something hard to find among recently touted AI triumphs in math.

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“This one is a bit different because people did look at it, and the humans that looked at it just collectively made a slight wrong turn at move one,” says Terence Tao, a mathematician at the University of California, Los Angeles, who has become a prominent scorekeeper for AI’s push into his field. “What’s beginning to emerge is that the problem was maybe easier than expected, and it was like there was some kind of mental block.”

The question Price solved—or prompted ChatGPT to solve—concerns special sets of whole numbers, where no number in the set can be evenly divided by any other. Erdős called these “primitive sets” because of their connection to similarly indivisible prime numbers.

“A number is prime if it has no other divisors, and this is kind of generalizing that definition from an individual number to a collection of numbers,” says Jared Lichtman, a mathematician at Stanford University. Any set of prime numbers is automatically primitive, because primes have no factors (except themselves and the number one).

Erdős also came up with the Erdős sum, a “score” you can calculate for any primitive set. He showed that the biggest the sum could be was about 1.6—and conjectured that this value must also hold for the (infinite) set of all prime numbers. Lichtman proved Erdős right as part of his doctoral thesis in 2022.

Erdős also noticed that the score drops if all of a set’s numbers are large—the larger the numbers, the lower the score. He guessed that the lowest this score could be was exactly one, a limit that the score would approach as the set’s numbers approached infinity. Lichtman tried to prove this, too, but got stuck like everyone else before him.

Price wasn’t aware of this history when he entered the problem into ChatGPT on an idle Monday afternoon. “I didn’t know what the problem was—I was just doing Erdős problems as I do sometimes, giving them to the AI and seeing what it can come up with,” he says. “And it came up with what looked like a right solution.”

He sent it to his occasional collaborator Kevin Barreto, a second-year undergraduate in mathematics at the University of Cambridge. The duo had jump-started the AI-for-Erdős craze late last year by prompting a free version of ChatGPT with open problems chosen at random from the Erdős problems website. (An AI researcher subsequently gifted them each a ChatGPT Pro subscription to encourage their “vibe mathing.”)

Reviewing Price’s message, Barreto realized what they had was special, and experts whom he notified quickly took notice.

“There was kind of a standard sequence of moves that everyone who worked on the problem previously started by doing,” Tao says. The LLM took an entirely different route, using a formula that was well known in related parts of math, but which no one had thought to apply to this type of question.

“The raw output of ChatGPT’s proof was actually quite poor. So it required an expert to kind of sift through and actually understand what it was trying to say,” Lichtman says. But now he and Tao have shortened the proof so that it better distills the LLM’s key insight.

More importantly, they already see other potential applications of the AI’s cognitive leap. “We have discovered a new way to think about large numbers and their anatomy,” Tao says. “It’s a nice achievement. I think the jury is still out on the long-term significance.”

Lichtman is hopeful because ChatGPT’s discovery validates a sense he’s had since graduate school. “I had the intuition that these problems were kind of clustered together and they had some kind of unifying feel to them,” he says. “And this new method is really confirming that intuition.”