How a 23‑Year‑Old Outsider Cracked a 60‑Year‑Old Math Conjecture Using ChatGPT

A 23‑year‑old amateur named Liam Price, who has no formal higher‑math background, partnered with Cambridge sophomore Kevin Barreto and fed a randomly selected unsolved Erdős problem (#1196) from the Erdős Problems website to ChatGPT.

The problem concerns primitive sets : a set of positive integers where no element divides another. For example, {2, 3, 7, 11} is primitive, while {2, 3, 7, 12} is not because 12 is divisible by 2 and 3. In 1968 Erdős, Sárközy, and Szemerédi conjectured that a certain sum over primitive sets has a clear asymptotic upper bound. After 60 years of effort, the best known bound was about 1.399, achieved by Jared Lichtman after seven years of work.

Traditional mathematicians approached the conjecture by translating the number‑theoretic question into probability‑theoretic language and following the “Mertens‑theorem” path, a standard route that generations of graduate students have taken. This conventional path, while natural, locked researchers into a narrow line of reasoning.

Price and Barreto instead used a simple prompt for GPT‑5.4 Pro. In roughly 80 minutes the model produced a proof that the sum is bounded by 1 + O(1/ log x) . The AI’s reasoning combined a Markov‑chain method with the von Mangoldt function , tools well‑known in other branches of analytic number theory but never before applied to the primitive‑set problem.

The raw output from GPT‑5.4 Pro was lengthy, tangled, and contained many logical jumps. Experts, including Barreto and other mathematicians, had to sift through the derivation to isolate the crucial new insight. Lichtman later remarked that the proof required expert filtering to be understood.

Fields Medalist Terence Tao praised the result, saying it was the first AI‑generated work reaching the level of Erdős’s “book of proofs.” He noted that humans had collectively taken a wrong first step for decades, whereas the LLM followed an entirely different route, revealing a new way of thinking about large integers and their structure.

The episode illustrates how an AI’s lack of historical baggage can become a structural advantage, allowing it to explore problem spaces that seasoned researchers avoid. It suggests that large language models may open new avenues in mathematical research, complementing traditional analytic techniques.

References: Scientific American article “Amateur Armed with ChatGPT Vibe Maths: A 60‑Year‑Old Problem” and a related X post (https://x.com/Ananyo/status/2047992864118894954).