The Erdős Proof

The Problem

A primitive set is a collection of whole numbers where no number divides any other. The primes are the simplest example: no prime divides another prime. But there are other primitive sets. {6, 10, 15} qualifies because none of those numbers divides the others. Erdős conjectured that for a specific mathematical score calculated from any primitive set, the primes achieve the minimum value, and that minimum approaches exactly 1 as the numbers grow. The conjecture had been open since the 1960s. Jared Lichtman, now at Stanford, proved that the primes are extremal among primitive sets as part of his doctoral thesis in 2022. Whether his result fully resolved the conjecture or left the limiting value open is a matter of mathematical interpretation. The problem Price and ChatGPT addressed had not been considered settled by the broader community. Price did not know any of this. He was doing Erdős problems recreationally, as he sometimes does, and fed one to ChatGPT to see what it would produce. His collaborator, Kevin Barreto, a second-year undergraduate at Cambridge, recognized the output might be significant and contacted the right people.

What the AI Did

ChatGPT, running GPT-5.4 Pro, spent 80 minutes and 17 seconds on the problem. It produced something that was, in Lichtman's words, "actually quite poor." The raw output was rough, incomplete, and would not have passed peer review in its generated form. But buried in the noise was a genuine insight. The AI used a well-known formula from analytic number theory, the von Mangoldt function, in a way that no previous researcher had thought to apply to primitive sets. There is a moment in the reasoning trace where the approach shifts without explanation. One technique ends. Another begins. No derivation connects them. Terence Tao, the UCLA mathematician and Fields Medalist who verified the work, explained what happened: "This one is a bit different because people did look at it, and the humans that looked at it just collectively made a slight wrong turn at move one." Every mathematician who had attempted the conjecture started with the same standard approach. They followed the conventional sequence of moves because that is what mathematicians do. The AI did not know the convention. It tried something else.

The Verification

Lichtman and Tao took the AI's output and extracted the valid proof from within it. They shortened it. They clarified it. They filled in the gaps the AI had left. The result is a proof that works and that opens a new way of thinking about primitive sets. Lichtman: "I had the intuition that these problems were kind of clustered together and they had some kind of unifying feel to them." The AI found the connection he had sensed but not formalized. Tao: "We have discovered a new way to think about large numbers and their anatomy. It's a nice achievement." But also: "I think the jury is still out on the long-term significance." The chain of contribution was precise. Price chose the problem and wrote the prompt. The AI generated a novel but messy approach. Barreto recognized the output might matter and routed it to experts. Tao and Lichtman verified, refined, and extracted the real insight. No single participant could have done this alone.

The Centaur Returns

In 1998, Garry Kasparov proposed that human-computer teams playing chess together could beat both humans and computers playing alone. He called them centaurs. For about a decade, he was right. The centaur period in chess produced some of the most creative play in the game's history. We wrote about this in The Centaur Problem. The chess centaur eventually died because the computers got good enough that humans just got in the way. Software development may be on the same clock. The Erdős proof suggests that in mathematics, the centaur is not dead. It is being born. The AI's contribution was novelty, not correctness. It could not produce a clean proof. It could not verify its own work. It could not explain why its approach was valid. What it could do was try something that no human would have tried, because every human who looked at the problem was constrained by the conventions of their field. This is what language models are genuinely good at. They have ingested every field simultaneously. They do not know which approaches are conventional and which are forbidden. They do not know which techniques belong to which subdomain. This ignorance, in the right context, is a form of freedom.

The Question

The optimistic reading of this story is that AI can accelerate mathematical discovery by cross-pollinating techniques across subfields that human specialists keep siloed. Tao believes the connection the AI found may have broader applications. The skeptics on Hacker News raised fair points. How many people fed Erdős problems to ChatGPT and got nothing? The article exists because this one worked. Survivorship bias is real. One commenter calculated the cost: billions of dollars went into building ChatGPT. Would funding mathematicians directly have produced the same result? Another noted that multiple models, including DeepSeek V4, produced similar proofs when given extended thinking time. The approach may have been discoverable by any sufficiently large model, which means it was on the edge of human reach. But the most important question is not whether AI can generate novel approaches. It clearly can. The question is whether every field has a Terence Tao waiting to catch the signal. Price's casual Monday afternoon prompt produced a proof that two world-class mathematicians found genuine and valuable. It worked because those two mathematicians existed, were available, took the output seriously, and had the training to distinguish signal from noise. Mathematics has Tao and Lichtman. The Erdős Problems wiki itself warns that AI solutions "lack additional context" and do not automatically merit publication. Verifying AI-generated proofs requires "mathematical maturity equivalent to understanding the original proof." You still need the humans. The centaur only works when both halves show up.

What This Means

This is a good story. A 23-year-old with no training, armed with curiosity and an AI, contributed to mathematics. A Fields Medalist took the work seriously and found it had value. The AI did something humans collectively could not do, not because it was smarter, but because it was unconstrained by convention. The wonder here is in the collaboration pattern, not in the technology alone. The AI was the instrument. The humans were the orchestra. Neither would have produced this music alone. For the dozens of fields where AI is being deployed to generate solutions, the Erdős proof is both a promise and a warning. The promise: AI can find things we would never find on our own. The warning: someone has to be able to tell when it does.