According to a report by Scientific American on April 24, Liam Price, a 23-year-old amateur mathematics enthusiast, used the latest large language model available in ChatGPT Pro to unexpectedly solve an Erdesh problem that has troubled the mathematics community for about 60 years without receiving advanced mathematical system training. This progress has attracted great attention from many well-known mathematicians.
According to reports, this achievement is of particular importance not only because the related problems have eluded many top mathematicians for a long time, but also because the proof idea given by AI is not a simple restatement of existing routines, but introduces a method that no one had thought could be used for such problems before.
The problem being solved this time discusses a special set of integers called "primitive sets". The so-called primitive set means that in the same set, no number is divisible by another number; in this sense, it extends the property of "prime numbers cannot be subdivided" from a single number to the entire set of numbers. Therefore, it is closely related to prime numbers, and any set of prime numbers naturally belongs to the original set.
The legendary Hungarian mathematician Paul Erdos once defined an "Erdös sum" for this type of primitive set, which can be understood as an indicator to measure a certain "weight" or "score" of the set. He has previously proved that the maximum value of this sum is approximately 1.6, and conjectured that the infinite set of all prime numbers also reaches this upper bound; Stanford University mathematician Jared Lichtman has proved this conjecture in his doctoral thesis in 2022. But a more difficult related conjecture is: when the numbers in an original set become very large, its "score" will continue to decrease, and its theoretical minimum limit should be exactly 1. In other words, what this question wants to prove is that as the set elements tend to infinity, this score will approach 1, and 1 is the lower bound that cannot be lower.
The report pointed out that Lichtman himself also tried to prove this conjecture, but failed like other previous researchers. Price said that he initially did not understand the ins and outs of this question. On an ordinary Monday afternoon, he casually entered Erdesh's question into ChatGPT as usual to see if the model could give ideas. As a result, the AI returned an answer that "seemed to be the correct answer."
Price then sent the results to his partner, Kevin Barreto, a second-year mathematics undergraduate at the University of Cambridge. The two had already attracted attention for randomly feeding public Erdesh puzzles to ChatGPT, and an AI researcher later even gave them a ChatGPT Pro subscription to support their experimental "atmospheric mathematics" attempt. After reviewing the results, Barreto realized that something was unusual and then notified relevant experts, who quickly responded.
Terence Tao, a mathematician at the University of California, Los Angeles, said that people who studied this problem in the past almost always followed a relatively standard starting path to carry out the derivation, but this time the large language model took a completely different route. According to the report, the AI used a formula that has long been known in related branches of mathematics, but no one had ever thought of applying it to this kind of problem. Tao Zhexuan believes that this shows that human researchers may collectively have some kind of "thinking bias" in their initial choice of direction, thus missing an actually more direct breakthrough path.
However, experts also emphasized that the proof text initially output by ChatGPT itself was not mature. Lichtman said that the quality of the original output is actually "pretty poor" and must be sorted out, screened and rewritten by professional mathematicians to truly understand the core logic it wants to express. Currently, he and Terence Tao have compressed and compiled this proof into a clearer version to more accurately extract key insights in the AI solution.
Rather than "this problem has been solved" itself, the mathematical community values more that this time AI seems to have opened up a new channel of thinking. Tao Zhexuan said that this work may mean that researchers have discovered a new way to understand "large numbers and their internal structures", and this connection may be transferred to a wider range of problems in the future; however, the long-term significance of this breakthrough still needs time to be tested. Lichtman believes that this result confirms his intuition since his graduate school days - there may be some common structure between many related problems, and the new method proposed by ChatGPT this time provides new evidence for this unity.