Xin Zhi Yuan Report
Editor: Taozi
[Xin Zhi Yuan Introduction] Seven years of professional research lost to a single instance of "vibe mathing." A 23-year-old with no background in advanced mathematics used a simple prompt to make ChatGPT crack a conjecture that has troubled humanity for 60 years in just 80 minutes. Terence Tao admits: We took the wrong path from the very first step.
A "century conjecture" that has plagued the mathematical community for 60 years has actually been cracked by an amateur!
He is only 23 years old, has never received any formal training in advanced mathematics, and relied solely on a single prompt to make ChatGPT solve this daunting problem.
After reviewing the proof, Terence Tao had only one thing to say—
For the past 60 years, humans have looked at this problem, and everyone collectively went down the wrong path at the very first step.
23-Year-Old Amateur Leaves the Internet Stunned
The protagonist of this story is named Liam Price.
He does not come from a "mathematical background," and his resume lacks any advanced mathematics degrees.
However, at the end of 2025, he teamed up with Kevin Barreto, a sophomore in the Mathematics Department at the University of Cambridge, to launch a nearly "crazy" experiment:
Randomly selecting unsolved problems from the famous Erdős Problems website and throwing them directly at ChatGPT.
No preliminary research, no reading of related papers, no starting from a specific analytical framework.
Just relying on intuition, describing the problem in the simplest language, and letting the large model find its own way.
The community coined a term for this method: "vibe mathing."
Before #1196, Price and Barreto had already used similar methods to make progress on several smaller problems, gradually attracting some attention.
When OpenAI heard about this, they gave the duo ChatGPT Pro subscriptions to encourage them to keep digging.
This move was later proven to be the highest-return investment in the history of mathematics in 2026.
But no one expected the real big fish to bite so quickly.
This time, they set their sights on Erdős Problem #1196, which concerns "primitive sets": a set where no two elements divide each other.
60-Year-Old Conjecture Proven in Just 80 Minutes by ChatGPT
The human mathematician who had made the most progress on this problem was Jared Lichtman from the University of Oxford.
He toiled over the primitive set problem for a full seven years, publishing multiple important papers and pushing the known upper bound step by step to approximately 1.399.
It seemed only one final step was needed to complete the proof. But that "final step" couldn't be taken for 7 years.
Unexpectedly, after Price sent the prompt, GPT-5.4 Pro reasoned for 80 minutes and delivered an asymptote of 1+O(1/log x), cutting straight to the core.
First, let's clarify the problem itself.
A "primitive set" is a set of positive integers where no number is divisible by another.
For example, in {2, 3, 7, 12}, 12 is divisible by 2 and 3, so it is not a primitive set, whereas {2, 3, 7, 11} is.
In 1968, Erdős and his collaborators Sárközy and Szemerédi proposed a conjecture: regarding a specific summation formula for primitive sets, there exists a clear upper bound in an asymptotic sense.
A concise formulation, yet a 58-year deadlock.
What's more crucial is not the difference in speed, but the difference in approach. All previous mathematicians who studied this problem, including Lichtman, defaulted to starting with the toolkit of analytic number theory.
This path seemed natural and has been walked for decades, but it locked the thinking into a narrow channel.
GPT-5.4 Pro took a completely different route: using a Markov chain method combined with von Mangoldt weights.
Both of these are mature tools in other branches of number theory, but no one had ever thought to apply them to the primitive set problem.
Intriguingly, Price admitted in an interview with Scientific American that GPT's raw output was "actually quite poor."
The proof was verbose, messy, and filled with logical leaps. It was Barreto and the experts who later got involved who identified that crucial, brand-new insight from a pile of chaotic derivations.
Lichtman's assessment was measured but carried immense weight: "This required an expert to sift through to truly understand what it was trying to express."
Then he said something that quieted the entire circle: "This is the first AI mathematical result to reach the level of The Book."
Anyone familiar with mathematics will immediately grasp the magnitude of this statement. "The Book" was a concept Erdős used during his lifetime: God has a book that contains the most elegant proof for every mathematical theorem.
Lichtman's point was that the AI not only solved the problem, but the solution itself was beautiful.
Terence Tao: Humanity Collectively Went Down the Wrong Path
The commentary from Fields Medalist Terence Tao provoked deep reflection from everyone.
Here is what he said—
People who studied this problem before often adopted a standard set of approaches at the very beginning.
The LLM, however, took a completely different route, using a formula that is well-known in related mathematical branches but had never been thought of being applied to this kind of problem.
This "collectively misguided first step" was the standard path formed since 1935:
Translating number theory problems into probability theory, going down the "Mertens' theorems" line—everyone assumed this was the right way.
Generation after generation of graduate students came in, learned this translation method first, and then added details on top of it.
GPT-5.4 Pro had never learned this "tradition." It turned around and used the von Mangoldt function—an object in analytic number theory that encodes the fundamental theorem of arithmetic—taking a completely different path.
Lichtman later explained: This formula is actually well-known to everyone in related mathematical fields, but no one ever thought to apply it to this Erdős problem.
Terence Tao's characterization of this result was even more profound: "We have discovered a completely new way of thinking about large integers and their structures."
The person who spent 7 years researching Lichtman's problem lost to a layman who didn't know how this problem "should be studied."
"Ignorance" has become a structural advantage in the AI era; without historical baggage, one naturally doesn't follow the collective down the wrong path.
The Keys to Mathematics Are Changing Hands
In 1900, David Hilbert posed 23 problems at the International Congress of Mathematicians in Paris, defining the direction of mathematics for the entire 20th century.
In that era, there were no more than a few hundred people worldwide who could touch the frontiers of mathematics.
On a Monday afternoon in April 2026, a 23-year-old, a single prompt, 80 minutes.
The doors of mathematics have not lowered their threshold, but there is a new key on the door.
The person holding this key doesn't need to spend ten years learning all the detours their predecessors took.
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