An AI built by OpenAI may have taken the closest step yet toward one of mathematics’ oldest unsolved problems – then the headlines went further than the math did.
The Riemann hypothesis has sat open since 1859. It is one of seven problems the Clay Mathematics Institute put a prize on. No one has ever proved it. What OpenAI released on October 6, 2026 was not a proof of it.
What the AI actually produced were two separate papers making a weaker claim. The two papers differ in length, in what they assert, and in whether anyone has formally checked them. One was quietly edited by a human before release. OpenAI has not said who did that or why.
Jared Duker Lichtman of Stanford University still called it the single most important advance AI has made in mathematics, and by a considerable distance. That praise and the caveats both belong in the same sentence.
Read to the end and you will know what each paper actually claims, which one has been independently checked, why mathematicians are reading the shorter and weaker version, and what questions the field still cannot answer.
| Feature | 7/8 version | 11/12 version |
|---|---|---|
| What it claims | All nontrivial zeros lie between real parts 1/8 and 7/8 | All nontrivial zeros lie between real parts 1/12 and 11/12 |
| Which is the tighter bound? | Yes — narrower strip, stronger result | No — wider strip, weaker result |
| Page count | 199 pages | 49 pages |
| Date | September 30, 2026 | Not separately dated |
| Formally verified? | Yes — verified in Lean proof assistant | No — not formally verified |
| Human-edited before release? | Not reported | Yes — the only paper among 722 to receive this treatment |
| Which version mathematicians are studying? | Not the focus | Primary focus of community review |
AI Did Not Prove the Riemann Hypothesis
OpenAI’s AI did not solve the Riemann hypothesis – the full 160-year-old conjecture remains entirely open.
What the full hypothesis actually demands is strict: every nontrivial zero of the Riemann zeta function must sit exactly on one specific line, the line where the real part of the input equals 1/2. That line runs through the middle of a vertical strip in the complex plane. The hypothesis says no zero can be anywhere else in that strip – not slightly to the left of center, not slightly to the right. One precise location, for infinitely many zeros.
What the AI produced is narrower than that.
Both papers confine the zeros to a band inside the strip but do not pull them all the way to the 1/2 line. That is a weaker statement – meaningful, but genuinely different from what the headline implied. As of their release, neither paper had cleared peer review; both were preprints, which means posted drafts awaiting independent scrutiny.
The mathematician Jared Duker Lichtman of Stanford University judged the result to be the greatest contribution AI has yet made to mathematics, and not by a small margin. That is a serious assessment from someone qualified to give it. It is also an assessment that stops well short of claiming the main problem is closed.
The reader who arrived here wondering whether the Riemann hypothesis is finally solved can set that question aside: it is not.
Two Papers Made Two Different Claims
What OpenAI released was not one result but two, and they are not the same claim.
The longer paper carries a September 30, 2026 date and runs to 199 pages. It argues that every nontrivial zero lies between real parts 1/8 and 7/8. A team then ran the argument through Lean, a proof-checking program, and it passed – making this the formally verified version. The shorter paper, 49 pages, pushes the boundary outward: it says zeros are confined between real parts 1/12 and 11/12. That second paper has not been put through any formal check.
The naming trips people up.
The 7/8 paper actually proves the tighter result. A strip from 1/8 to 7/8 is narrower – closer to that target line at 1/2 – than a strip from 1/12 to 11/12. Bigger fractions in the name, smaller gap from center. Reading the numbers without pausing on what they represent gives the wrong picture.
Both papers arrived on October 6, 2026, among 722 total papers OpenAI posted that day. The quasi-RH work was flagged as the standout among the 377 announced results that day. Most coverage folded the two versions into a single claim – but they differ in what they assert, how long they run, and whether any independent check exists. The table in this piece lays those differences out side by side, because collapsing them loses the most important detail: the stronger, verified result and the weaker, unverified result are doing different things, and the field is treating them very differently.
The Weaker Paper Gets More Attention Because It Is Shorter
The 7/8 paper is stronger, formally verified, and largely being set aside. The reason is length.
A 199-page proof requires a sustained, coordinated effort to check by hand. A 49-page proof is something a single mathematician can sit down with over days or weeks. Maksym Radziwiłł of New York University put it plainly: the reason people are looking at the 11/12 result is that it is shorter. That is the whole explanation.
So the paper drawing active mathematical attention is the one with no formal verification.
The Lean check covers only the 7/8 result – the one that is sitting largely unread. The 11/12 paper, which is the object of scrutiny, has no comparable mechanical check behind it. The field is, in practice, studying the weaker and unverified version because it is the one human readers can get through.
That asymmetry carries an uncomfortable implication. Formal verification is often described as the gold standard for machine-generated proofs – a way to trust a result that no human could fully audit by hand. Here, the formally verified paper is the one nobody is checking, and the unverified paper is the one everyone is working through. What that says about the practical value of formal verification in a case like this is a question the field does not yet have a settled answer to.
The two papers are doing different work in how mathematicians are processing this result. Collapsing them into a single claim erases the most live tension in the story.
⚠️ COMMON MISTAKE
Stronger Bound Does Not Mean Bigger Number
The 7/8 result is stronger than the 11/12 result, even though 7/8 is numerically larger. A strip from 1/8 to 7/8 is narrower — it squeezes the zeros closer to the 1/2 target line. The naming is genuinely counterintuitive, and most coverage gets it backwards.
The Hypothesis Has Resisted Proof Since 1859
The Riemann hypothesis is a statement about where certain special numbers live – and it has defeated every mathematician who has tried to prove it for more than 160 years.
To picture the problem: the Riemann zeta function produces outputs of zero at certain input values. Those inputs are called nontrivial zeros. They all land inside a vertical strip in the complex plane – a region where the real part of the input runs between 0 and 1. Bernhard Riemann made a sharper claim: every single one of those zeros sits exactly on the center line of that strip, where the real part equals 1/2.
That conjecture has never been proved.
Progress did come, though – just short of the full claim. In 1896, two mathematicians showed the zeros cannot crowd against the strip’s outer edges, where the real part equals 0 or 1. That narrower result was enough to prove the prime number theorem, which describes the average rate at which prime numbers thin out as numbers grow larger. A weaker boundary condition still carried enormous consequences.
The stakes for the full conjecture are hard to overstate. Enrico Bombieri put it plainly: if the Riemann hypothesis turned out to be false, it would throw the distribution of prime numbers into disarray. The hypothesis is the only problem to appear on both David Hilbert’s famous list of open questions and the Clay Mathematics Institute’s Millennium Prize list. More than a century of effort by some of the best mathematical minds alive has produced deeper partial results, but the full proof has not arrived. That long trail is the measure against which any new claim has to be read.
The Scale of the Problem — and the Result
Quasi-RH Already Delivers Most of What the Full Proof Would
The quasi version of the hypothesis does real mathematical work, even though it stops short of the full conjecture.
Maksym Radziwiłł of New York University is direct about this: almost everything that mathematicians care about as a consequence of RH can already be derived from quasi-RH alone. That is a strong claim from a mathematician who has spent years working on exactly these problems.
The 1896 result shows why this kind of logic holds. Those two mathematicians only proved a narrow zero-free region near the strip’s edges – not the precise 1/2-line claim at all. Yet that modest improvement in where the zeros could sit was enough to pull the prime number theorem across the finish line. A tighter bound on zeros feeds into sharper statements about how primes spread out, even if the tightest possible bound remains unproved.
Prime distribution is not only a pure math puzzle. The security behind internet communication – every password, every bank transaction – depends on how hard it is to factor very large numbers built from primes. The mathematical ground that security rests on runs directly through results about how primes behave at large scales.
Radziwiłł’s assessment is a logical statement about what follows from the quasi result, not a claim that the full conjecture is now unnecessary. The full problem still stands open, and mathematicians still want to solve it. But the quasi-RH result carries genuine weight on its own terms – most of the downstream mathematical consequences of the full proof would follow from it, and that is why the result commands serious attention even with the headline’s overpromise stripped away.
Checking 12 Trillion Zeros Is Not the Same as Proving It
Twelve trillion confirmed zeros sounds like near-certainty – it is not, and that gap is exactly why a proof still matters.
Computers have checked that the first 12 trillion nontrivial zeros each sit exactly on the 1/2 line. That is an enormous number. Dave Platt and Tim Trudgian pushed the formal frontier to all zeros up to 3×10^12, published in the Bulletin of the London Mathematical Society. That was the state of play before the AI papers arrived.
None of it constitutes a proof.
The zeta function produces infinitely many nontrivial zeros. One single exception – anywhere in that infinite list – would be enough to disprove the hypothesis entirely. No finite check, however large, can rule out an exception at some unreachably large value nobody has computed yet. Checking a range and proving a rule are two entirely different operations.
Think of it this way: someone could confirm that every even number they have ever tested is divisible by two. That confirms a pattern. It does not prove the pattern holds forever – a separate logical argument does that. The same structure applies here.
This is precisely where the quasi-RH result adds something numerical work never could. A proof – even a weaker one covering a range of possible locations rather than the exact 1/2 line – makes a statement about all zeros at once, not just the ones a computer has reached. The 12 trillion figure is striking context; it is not a substitute for what a proof actually does.
The Proof Connected Known Ideas Rather Than Inventing New Ones
What the AI did was not discover new mathematics – it found a connection between pieces that already existed.
A researcher at the Georgia Institute of Technology named Alexander Dunn described the approach as a surprising synthesis of well-known ideas. No powerful new machinery, no novel framework that mathematicians had never seen. The proof assembled existing techniques in a way nobody had managed before – and that is the achievement, if it holds up.
One detail makes the situation stranger than it sounds. Some of those existing techniques were developed by Dunn himself, and by Maksym Radziwiłł of New York University. Both are now among the mathematicians working to understand a proof that draws on their own prior results. They are, in a sense, reviewing work built partly from their own toolkit.
That is not a conflict – it is a precise illustration of where AI sits relative to the field right now. It did not outpace the researchers working at the frontier; it combined what they had already built. AI synthesizing existing human work mirrors how AI accelerated vaccine research by recombining known biological data rather than inventing new science.
The reader assumption that a significant AI result must involve new concepts does not hold here. Finding a synthesis nobody saw is a real contribution. The components existed; the combination, apparently, did not. Whether that combination is correct is a different question entirely – and it is the one the field has not yet answered.
Both Papers Fall Far Short of Publication Standards
Whether the synthesis is correct, nobody can say with confidence yet – and part of the reason is that the papers are genuinely hard to read.
Radziwiłł was direct about the writing. He described the papers as dreadfully composed – so badly that any mathematical journal would reject them outright – and said they fell well below the norms of clarity and presentation that the field expects. That is a working mathematician describing texts he is actively trying to assess.
Poor prose does not equal wrong mathematics. But it does mean reviewers cannot easily tell a valid step from an unclear one – and that slows everything down.
The verification picture is asymmetric in a way that compounds the problem. The 7/8 paper – 199 pages, the stronger bound – was formally checked in Lean, a proof-assistant program that flags logical errors. The 11/12 paper, the shorter one that mathematicians are actually studying, has no such check at all. The paper receiving human attention is the one with no machine safety net.
Mathematicians are still working through what both proofs contain. Saying the results are unconfirmed is not caution for its own sake – it is an accurate description of where things stand. Possibly correct, possibly significant, and not yet independently verified: that is the precise position the field is in.
One Paper Was Human-Edited; OpenAI Has Not Said Why
Provenance trouble does not stop at writing quality. Among all 722 papers OpenAI published on October 6, 2026, exactly one was touched by a human hand before release – and that one is the 11/12 paper, the shorter result that every working mathematician is currently trying to read.
OpenAI has not said who that person was, what they changed, or why this paper alone warranted the treatment. The company’s only public statement on authorship credits the work collectively to the entire organization. That sentence raises more questions than it answers.
It is not an accusation of misconduct. It is a factual gap, sitting in plain view. Anyone outside the company who wants to know what a human contributor added – whether a line of reasoning, a structural choice, a correction – simply cannot find out. The record does not exist, or has not been shared.
That gap matters most precisely because of which paper it concerns. The 11/12 result is the one mathematicians are spending their time on. Its significance as an AI achievement depends, at least in part, on understanding what AI actually produced. Right now, that understanding is unavailable – and the paper drawing the most scrutiny is the one whose origins are least clear.
💡 PRO TIP
What Lean Verification Actually Guarantees
Lean is a proof assistant: software that checks each logical step against formal rules. Lean verification of the 7/8 paper means no step in that proof violates the formal logic the system checks — it does not mean mathematicians have read and understood the argument. The two are genuinely different kinds of confidence.
Prime Numbers Underlie the Encryption Everyone Uses Daily
The reason this result reaches beyond mathematics comes down to prime numbers and what they protect.
Every time you enter a password or send a payment online, the security holding that data together rests on a hard fact about primes: multiplying two large primes together is easy, but working backwards to find those primes from the result is extremely difficult. That difficulty is the lock. Modern encryption systems are built on it.
How primes are spread among large numbers determines how strong that lock can be. The prime number theorem, proved in 1896, gave the first real formula for that spread. Radziwiłł has noted that quasi-RH delivers nearly all the same practical consequences for prime distribution that a full proof of the Riemann hypothesis would bring – meaning the gap between the two results, in applied terms, is smaller than it might appear.
That is a direct line from a short AI preprint to the math behind internet security.
The connection is real, but the scale matters. This result does not break any encryption. It does not call for changes to any security system in use today. What it does is sharpen the mathematical picture of how primes thin out, and that picture is the ground everything else is built on.
The stakes are long-run and foundational – not something a user needs to act on now, but not abstract either. Results at this level feed into the deep structure that security researchers and mathematicians rely on for decades to come.
Machine-Generated Proofs Force New Questions About Mathematical Review
These two papers also put pressure on something mathematics has never had to settle before: what counts as a proof when a machine wrote it.
The 7/8 paper was checked by the Lean proof assistant – a formal verification that went through every logical step. Mathematicians are not studying it. The 11/12 paper has no such check – and that is the one people are actually reading. The community is spending its attention on the unverified result and largely setting aside the verified one, because 49 pages is manageable and 199 pages is not.
That is a strange place to be.
Both papers could, in principle, contain correct mathematics and still be impossible to publish right now. Radziwiłł was direct: he described the writing as dreadful enough to guarantee rejection at any mathematics journal, and said the texts sit far below the norms of clarity and exposition that the field expects. Bad writing does not make the steps wrong. But when a reviewer cannot follow the argument, checking it takes far longer – and the difference between a clear proof and an opaque one is not a style preference, it is a practical barrier to knowing whether the result is true.
No agreed process exists yet for handling this kind of work. When a human mathematician submits a proof, the field has well-worn ways to check it, credit it, and publish it. A 199-page machine-generated text, formally verified but unreadable, does not fit that process. Neither does a 49-page version that was human-edited under circumstances nobody has explained.
Mathematicians are debating what to do. The debate is live and the rules are not written yet.
Whether This Opens a Path to the Full Proof Is Genuinely Unknown
Neither paper proves the full Riemann hypothesis, and nothing else has either – it remains exactly as open as it was on October 5, 2026.
The real question is whether the 11/12 result points in the direction of a full proof. If the 49-page paper turns out to be correct, does the method it uses suggest a route to showing all zeros lie on the 1/2 line? Or does the approach stop at the quasi version, with no bridge forward?
Nobody knows yet.
The people best placed to answer that are still reading the paper. They find it badly written. It has no formal verification. The person who edited it before release has not been named. Those are not reasons to dismiss the result – but they are real obstacles to understanding what the result actually contains, let alone where it might lead.
Premature conclusions cut both ways here. Writing the result off because it falls short of the full conjecture misses what quasi-RH already delivers. Treating it as a near-proof of the full hypothesis misreads what the papers claim. Mathematicians are still working out what the proof contains and what it might mean.
The honest answer is that the field does not yet know whether this is a step on a longer road or a genuine result that stops where it stops. That is not pessimism. It is where the mathematics actually sits right now.
Frequently Asked Questions
Did AI actually solve the Riemann hypothesis?
No. OpenAI’s AI produced two quasi-Riemann hypothesis results that narrow the strip where certain mathematical objects can live, but do not place them on the precise 1/2 line the full conjecture requires. The 160-year-old full conjecture remains entirely open and unproved.
What is the difference between the 7/8 and 11/12 quasi-RH results?
The 7/8 paper claims all nontrivial zeros lie between real parts 1/8 and 7/8 — a tighter, stronger bound — and has been formally verified in the Lean proof assistant. The 11/12 paper claims zeros lie between 1/12 and 11/12, a weaker result, and has not been formally verified. Despite being weaker, the 11/12 paper is receiving more attention because it is 49 pages rather than 199.
What does it mean that the 7/8 proof was verified in Lean?
Lean is proof-checking software that validates each logical step against formal rules. Lean verification means no step in the 7/8 paper violates formal logic — it does not mean mathematicians have read and understood the argument. Formal verification and human comprehension are two different kinds of confidence, and the field currently has only the former for the stronger result.
Why does the Riemann hypothesis matter for everyday life?
Prime numbers underlie the encryption that secures internet communication — every password and transaction depends on how hard it is to factor large numbers built from primes. Results about how primes distribute feed into the mathematical foundations of those systems. Quasi-RH advances that understanding at a foundational level, though it does not change how encryption works in practice today.
Why haven’t mathematicians checked 12 trillion zeros and called it proven?
The zeta function has infinitely many nontrivial zeros, and one exception anywhere disproves the hypothesis. Platt and Trudgian verified in a 2021 paper in the Bulletin of the London Mathematical Society that the hypothesis holds for all zeros up to 3×10^12 — but no finite count of confirmed cases, however large, constitutes a proof for all zeros. Numerical verification and mathematical proof are categorically different tools.
Why was only the 11/12 paper human-edited, and what did that editing involve?
The 11/12 paper is the only one among the 722 papers OpenAI released on October 6, 2026 that received human editing before release. OpenAI has not identified who edited it, what the edits involved, or why this paper alone received that treatment. The company’s public statement says only that the paper is the result of work from everyone at OpenAI, leaving the nature and extent of human contribution unclear.
What Confirming the 49-Page Proof Would — and Would Not — Settle
Here is the plain verdict. OpenAI’s AI produced two quasi-Riemann hypothesis results. They make different claims. Only one has been formally verified. Neither has been peer-reviewed. One was human-edited under circumstances OpenAI has not explained. Both are genuinely notable. Both are genuinely unresolved.
The headline you saw said the Riemann hypothesis was proved. What happened on October 6, 2026 is both more interesting and more ambiguous than that.
The question mathematicians are sitting with now is this: if the 49-page paper survives scrutiny, does the method it uses point in any direction toward the full conjecture – the line at 1/2 – or does it simply stop where it stops? No one yet knows. The full Riemann hypothesis is exactly as open today as it has ever been. That is not a failure. It is where the mathematics actually is.
Sources
- AI Proved the Quasi-Riemann Hypothesis. What Happens Now? | Quanta Magazine – Quanta Magazine (news report) · 2026-10-19
- Jared Duker Lichtman | Mathematics – mathematics.stanford.edu (press release)
- The Riemann hypothesis is true up to 3·1012 – Bulletin of the London Mathematical Society (peer-reviewed paper) · 2021-01-22
