23-Year-Old Outsider Uses ChatGPT to Solve 60-Year-Old Math Conjecture

A 23-year-old with no higher math training has solved a 60-year-old math conjecture using a simple prompt with ChatGPT.

After reviewing the proof, mathematician Terence Tao remarked that humanity had collectively gone astray from the start.

The Story of Liam Price

The protagonist of this story is Liam Price, who does not have a formal background in mathematics. In late 2025, he teamed up with Kevin Barreto, a second-year math student at Cambridge University, to embark on a seemingly “crazy” experiment: randomly selecting unsolved problems from the Erdős Problems website and presenting them to ChatGPT.

They approached the problems without prior research, reading related papers, or starting from any analytical framework. Instead, they relied on intuition, using simple language to describe the issues and letting the large model find its way. This method was dubbed “vibe mathing.”

Before tackling Erdős Problem #1196, Price and Barreto had made progress on several smaller problems using a similar approach, gaining some attention. OpenAI, upon hearing about their work, provided them with a ChatGPT Pro subscription to encourage further exploration, a decision that would prove to be one of the most rewarding investments in the history of mathematics.

The Breakthrough in 80 Minutes

The most advanced human mathematician on this problem was Jared Lichtman from Oxford University, who spent seven years working on it, publishing several important papers and pushing the known upper bound to about 1.399.

However, after Price sent out the prompt, GPT-5.4 Pro took just 80 minutes to provide an answer of asymptotic 1 + O(1/log x).

To clarify, a “primitive set” is a collection of positive integers where no element divides another. For example, {2, 3, 7, 12} is not a primitive set because 12 can be divided by 2 and 3, while {2, 3, 7, 11} is a primitive set.

In 1968, Erdős and his collaborators Sárközy and Szemerédi proposed a conjecture regarding a specific summation related to primitive sets, suggesting there exists a clear upper bound in asymptotic terms. This conjecture had been stagnant for 58 years.

The critical difference was not just the speed of the solution but the approach. All previous researchers, including Lichtman, had assumed a path based on analytic number theory tools. This route seemed natural and had been followed for decades, but it confined thinking to a narrow channel.

In contrast, GPT-5.4 Pro took a completely different path, employing Markov chain methods combined with von Mangoldt weights. These elements are established tools in other branches of number theory, but no one had thought to apply them to the primitive set problem.

Interestingly, Price admitted in an interview with Scientific American that GPT’s initial output was “actually of poor quality.” The proof was lengthy and chaotic, with logical jumps throughout. It was Barreto and later experts who identified the key new insight from a mass of disorganized derivations.

Lichtman provided a measured yet significant evaluation: “This requires experts to sift through to truly understand what it’s trying to express.”

He then made a statement that silenced the community: “This is the first AI mathematical achievement that reaches the level of Erdős’s book.”

Familiarity with mathematics allows one to appreciate the weight of this statement. Erdős’s book refers to a notion he proposed: that God has a book containing the most elegant proofs of every mathematical theorem. Lichtman implied that the AI not only solved the problem but that the solution itself was beautiful.

Terence Tao: A Collective Misstep

Fields Medalist Terence Tao’s commentary prompted deep reflection. He stated that previous researchers often began with a standard set of approaches, while LLMs took a completely different route, utilizing a formula well-known in related mathematical branches but never thought to apply it to this type of problem.

This “collective misstep” originated from a standard path established since 1935: translating number theory problems into probability theory, following the line of the Mertens theorem, with everyone assuming this route was correct. Generations of graduate students learned this translation method first before adding details.

GPT-5.4 Pro had never learned this “tradition.” Instead, it directly utilized the von Mangoldt function—an object encoding the fundamental theorem of arithmetic in analytic number theory—taking a completely different route.

Lichtman later explained that this formula was familiar to many in the relevant mathematical fields, but no one had thought to apply it to the Erdős problem. Tao characterized the result more harshly: “We have discovered a new way to think about large integers and their structures.”

A person who spent seven years studying Lichtman’s problem lost to an outsider who did not know how the problem “should be studied.” In the age of AI, “ignorance” has become a structural advantage, free from historical burdens, thus naturally avoiding collective missteps.

A New Key for Mathematics

In 1900, David Hilbert proposed 23 problems at the International Congress of Mathematicians in Paris, defining the direction of mathematics for the entire 20th century. At that time, only a few hundred people worldwide could touch the frontiers of mathematics.

On a Monday afternoon in April 2026, a 23-year-old, a simple prompt, and 80 minutes changed the landscape of mathematics. The gates of mathematics have not lowered their standards, but a new key has been added. Those holding this key do not need to spend a decade learning all the detours taken by their predecessors.