ChatGPT Helps Amateur Math Enthusiast Solve 60-Year-Old Erdős Discrepancy Problem
According to a report by Scientific American on April 24th, 23-year-old amateur math enthusiast Liam Price, without formal higher-level mathematical training, unexpectedly made progress in solving a problem posed by Erdős that has puzzled mathematicians for about 60 years, with the help of the latest large language model available in ChatGPT Pro. This development has attracted considerable attention from several prominent mathematicians.
The report states that this achievement is particularly noteworthy not only because the problem had stumped many top mathematicians for a long time, but also because the proof approach provided by AI was not a simple repetition of existing routines, but introduced a method previously not considered applicable to such problems.
The problem that has been solved concerns a special set of integers called “primitive sets.” A primitive set is one in which no number in the set is divisible by another; in this sense, it extends the property of “primes being indivisible” from individual numbers to the entire set of numbers, and is therefore closely related to primes. Any set of primes naturally belongs to a primitive set.
Hungarian legendary mathematician Paul Erdős defined an “Erdős sum” for these primitive sets, which can be understood as an index to measure the set’s “weight” or “score.” He previously proved that the maximum value of this sum is approximately 1.6, and conjectured that the infinite set consisting of all primes also reaches this upper bound exactly; Jared Lichtman, a mathematician at Stanford University, proved this conjecture in his 2022 doctoral dissertation. However, a more difficult related conjecture is: as the numbers in a primitive set become very large, its “score” will continue to decrease, and its theoretical lower limit should be exactly 1. In other words, the problem to be proved is that as the set elements tend to infinity, this fraction will approach 1, and 1 is precisely the lowest possible lower bound.
The report points out that Lichtman himself had tried to prove this conjecture, but like other researchers before him, he failed. Price said that he initially did not understand the background of the problem, but on a regular Monday afternoon, he casually entered the Erdős problem into ChatGPT as usual to see if the model could provide ideas, and the AI returned a “seemingly correct answer.”
Price then sent the result to his partner, Kevin Barreto, a sophomore mathematics student at the University of Cambridge. The two had previously attracted attention for randomly feeding open Erdős problems to ChatGPT. An AI researcher later even gifted them a ChatGPT Pro subscription to support their experimental “ambient mathematics” attempts. Barreto, after reviewing the result, realized that something was unusual and then notified relevant experts, and the academic community quickly responded.
Terence Tao, a mathematician at UCLA, said that people who have studied this problem in the past have almost always started their derivations along a relatively standard path, but this time the large language model took a completely different route. The report states that the AI called a formula already well-known in the relevant branch of mathematics, but no one had ever thought of applying it to this problem. Tao believes that this shows that human researchers may have collectively experienced a kind of “cognitive bias” in their initial direction selection, thus missing a more direct breakthrough path.
However, experts also emphasized that the original proof text output by ChatGPT was not mature. Lichtman said that the quality of the original output was actually “quite poor” and had to be sorted, filtered, and rewritten by professional mathematicians to truly understand the core logic it was trying to express. Currently, he and Tao have compressed and organized the proof into a clearer version to more accurately distill the key insights from the AI’s solution.
More than the fact that “the problem has been solved,” the mathematical community values the fact that AI seems to have opened a new avenue for thinking. Tao said that this work may mean that researchers have discovered a new way to understand “large numbers and their internal structure,” and this connection may be transferable to broader problems in the future; however, the long-term significance of this breakthrough still needs time to verify. Lichtman believes that this result confirms his intuition since his graduate studies—that there may be a common structure between many related problems, and the new method proposed by ChatGPT just now provides new evidence for this unity.