GoKawiilA ChatGPT AI has proved a conjecture with a method no human had thought of. Experts believe it may have further uses
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.
On supporting science journalism
If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
“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.
... continue reading