How a 23‑Year‑Old Non‑Mathematician and ChatGPT Cracked a 60‑Year‑Old Conjecture

Problem definition

A primitive set is a collection of positive integers in which no element divides another. The Erdős–Sárközy–Szemerédi conjecture (Problem #1196, posed in 1968) asserts that for any primitive set S the sum \sum_{n\in S}\frac{1}{n\log n} has an asymptotic upper bound of the form 1+O\bigl(1/\log x\bigr) as the largest element x of S tends to infinity.

Historical attempts

For six decades the standard analytic‑number‑theory approach translated the problem into a probabilistic framework via the Mertens theorem. The most advanced human effort was by Jared Lichtman (Oxford), who spent seven years refining the bound to approximately 1.399 using this traditional toolbox.

AI‑driven approach

In late 2025, Liam Price (no formal higher‑math training) and Cambridge sophomore Kevin Barreto submitted a concise natural‑language description of Problem #1196 to GPT‑5.4 Pro. Within 80 minutes the model generated a proof sketch that combined two previously unrelated tools:

A Markov‑chain framework for modeling the growth of primitive sets.

The von Mangoldt function \Lambda(n), which encodes the fundamental theorem of arithmetic.

This combination produced an asymptotic upper bound of the form 1+O\bigl(1/\log x\bigr), effectively improving the known bound from ~1.399 to the conjectured optimal constant.

Expert assessment

Price reported that the raw output was “very poor” and tangled, but Barreto and subsequent domain experts extracted the crucial insight. Lichtman remarked that the result required expert filtering to be understood, while Terence Tao noted that the community had collectively followed a misguided path for the past 90 years and that the AI‑generated proof represents a completely new way of thinking about large integers.

Significance

The AI method diverged from the traditional route of translating the problem into probability via the Mertens theorem. Instead, it applied the von Mangoldt function—well‑known in analytic number theory but never before used for primitive‑set problems—together with a Markov‑chain analysis. This demonstrates that large language models can discover novel mathematical pathways without prior exposure to the established literature.

Sources: Scientific American interview (2025); X post discussing the breakthrough (2025).