The 166-Year-Old Prime Number Puzzle an AI May Have Cracked — Almost
The Riemann hypothesis is arguably the biggest unsolved problem in mathematics. First put forward by Bernhard Riemann in 1859, it has resisted every attempt at a proof for more than 160 years. It even carries a $1 million prize from the Clay Mathematics Institute as one of its seven Millennium Prize Problems. On 6 October 2026, OpenAI announced that its AI had proved a weaker version of it — the so-called "quasi-Riemann hypothesis" — and mathematicians have been poring over the claim ever since.
Jared Duker Lichtman of Stanford University called it "the most significant AI-for-math achievement so far, by quite a margin." Maksym Radziwiłł of New York University said the implications were "staggering", adding: "In a sense, the vast majority of consequences of RH already follow from quasi-RH."
Primes, and a function that counts them
To understand the puzzle, start with prime numbers — numbers divisible only by 1 and themselves, like 2, 3, 5, 7 and 11. Mathematicians call them the "atoms of arithmetic", because every whole number can be built by multiplying primes together. As you move along the number line, primes thin out, and Riemann showed that exactly how they thin out is governed by a special function, the Riemann zeta function.
The Riemann hypothesis is about where this function equals zero. It says that all of its interesting zeros — the "nontrivial" ones — lie on a single vertical line, where the real part is exactly 1/2. Mathematicians have checked by computer that the first 12 trillion nontrivial zeros do sit on that line. But checking trillions of cases is not a proof: even one stray zero would, as the mathematician Enrico Bombieri wrote, "create havoc in the distribution of prime numbers."
The "quasi" version asks for less. Instead of proving that all the zeros sit exactly on the line, it asks only to prove that they all lie inside a narrower strip than the one already known. The narrower the strip, the more mathematicians can say about how the primes are distributed — and most of the practical consequences of the full hypothesis would follow.
Two versions, one machine-checked
OpenAI announced two versions of its proof. The first, dated 30 September 2026 and 199 pages long, claims that all nontrivial zeros lie between 1/8 and 7/8, and was formally checked using Lean, a programming language that verifies every logical step of a proof by computer. The second version, completed almost a week later and only 49 pages long, claims a weaker result — the zeros lie between 1/12 and 11/12 — but it has not yet been formally verified in Lean.
Both papers, mathematicians say, are badly written — Radziwiłł told Quanta Magazine they "fall short of the standard of exposition and clarity in our area by a wide margin" and would be rejected immediately by any maths journal. So experts have set the longer paper aside and focused on the shorter one, which seems to use similar techniques. Peter Sarnak, a leading number theorist at Princeton, has even repurposed a course he is teaching this semester to study it. Radziwiłł says he can now sketch the proof's arguments in half a page, and experts largely agree the result is correct.
What surprised mathematicians most is that the proof needed no new concepts at all. It was, said Alexander Dunn of the Georgia Institute of Technology, "a surprising synthesis of well-known ideas". The key was an obscure family of mathematical objects called metaplectic forms, which Dunn has spent five years studying, and a formula he and Radziwiłł proved in 2021 that turned out to be a crucial ingredient. For decades, number theorists had believed their tools were not strong enough to settle the quasi-Riemann hypothesis. The AI's "belief" that they might be enough was, Radziwiłł said, "a pretty bold belief".
Not a theorem yet
There is an important caution. Sarnak has said he will not be satisfied that it is "a Theorem" until humans have checked and understood every detail. The full Riemann hypothesis, meanwhile, still looks out of reach: as Will Sawin of Princeton put it, once mathematicians have fully absorbed the new methods, "we will still feel as far off from the full Riemann hypothesis as we were before." Thomas Bloom of the University of Manchester agreed — but added, "we would have said the same about quasi-RH a few weeks ago! At this point nothing would surprise me."
For students, this is a wonderful window into how mathematics really works: a 166-year-old question about the primes, a machine that produced a proof no human had found, and a community of experts now doing the slow, careful human work of checking it, line by line.
If your child loves puzzles, patterns or prime numbers, a first lesson in maths can start with this story — and with the primes themselves. Book your first lesson
Sources:
- Quanta Magazine, "AI Proved the Quasi-Riemann Hypothesis. What Happens Now?", 19 October 2026: https://www.quantamagazine.org/ai-proved-the-quasi-riemann-hypothesis-what-happens-now-20261019/
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