| Title | An Amateur Just Solved a 60-Year-Old Math Problem—by Asking AI | |
| Date | Monday May 04, @09:39AM | |
| Author | hubie | |
| Topic | ||
| from the dept. | ||
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.
"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 Duker 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).
[...] "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."
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