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OpenAI Math Proofs Explained: Has AI Solved Math Problems? What the 722 Manuscripts Really Show

OpenAI math proofs, explained: 722 AI manuscripts released on 6 October 2026. Which results are verified, which were withdrawn, and the lesson for your AI use.

AI & Automation · 30 Oct 2026 · 11 min read

OpenAI Math Proofs Explained: Has AI Solved Math Problems? What the 722 Manuscripts Really Show

Handwritten mathematical formulas in white chalk covering an old blackboard

By@ashtech52Lead Full-Stack Engineer

On 6 October 2026, OpenAI posted 722 maths manuscripts from an unreleased AI model to GitHub, claiming progress on 372 open problems. A day later it withdrew three over a sign error.

So has AI solved math problems that humans couldn't? Some, very probably. Most, nobody knows yet.

That gap between "claimed" and "checked" is the real story behind the OpenAI math proofs, and the business lesson: AI made answers cheap, and checking them is now the bottleneck.

If you're adding language models to your own workflows, that's the problem our LLM integration work is built around.

Has AI solved math problems? The short answer

Partly. OpenAI's unreleased model produced 722 manuscripts claiming progress on 372 open problems, and leading number theorists largely agree its quasi-Riemann result is correct. But only about 42% of the top-line results have machine-checked Lean proofs, three papers were withdrawn the next day, and as of 26 October 2026 most of the rest still await expert review.

  • Impressive: the quasi-Riemann result, which Stanford's Jared Duker Lichtman calls the most significant AI maths achievement so far.
  • Unverified: the results without Lean proofs are long English write-ups that experts have not finished reading.
  • Already corrected: three manuscripts withdrawn and 14 revised within 24 hours of release.
  • Not solved: the Riemann hypothesis itself. The Navier-Stokes claim from September is still being checked, and the Clay Mathematics Institute has not ruled on it.

Timeline: how OpenAI's maths claims built up in 2026

The October release didn't come out of nowhere. Here's the sequence, with the reaction at each step:

  1. October 2025: an OpenAI executive posts that GPT-5 found solutions to 10 open Erdős problems. The solutions were already in the literature, and the post was deleted.
  2. 20 May 2026: OpenAI says a reasoning model disproved a geometry conjecture Paul Erdős posed in 1946. Noga Alon, Melanie Wood and Thomas Bloom publish remarks supporting the disproof.
  3. Early August 2026: OpenAI publishes "Ten advances in mathematics and theoretical computer science", which it says come with Lean proofs. By 6 August, experts told Scientific American that two of the ten reused earlier published work without proper credit.
  4. 8 September: OpenAI says a multi-agent system, about 10,000 AI agents running for roughly 88 hours, proved that the Navier-Stokes equations of fluid flow can "blow up". It says it won't claim the $1 million Millennium Prize. NYU's Tristan Buckmaster says he and Levent Alpöge were working on a closely related approach; OpenAI denies using their work.
  5. 11 September: the Clay Mathematics Institute says the problem "has apparently been settled" and that its evaluation process is "deliberately unhurried".
  6. 21 September: an independent Advisory Group on Mathematics and Artificial Intelligence, hosted at the Institute for Advanced Study in Princeton, is announced with nine mathematicians. It advises; it has no decision-making power at any AI company.
  7. 6 October: the big release: 722 manuscripts in 372 result families, from an evaluation of about 4,000 problems, averaging about three hours of ChatGPT Pro thinking per result.
  8. 7 October: three manuscripts withdrawn and 14 revised. The Association for Human Mathematics calls the release "a demonstration of power".
  9. 8 October: Retraction Watch reports an OpenAI spokesperson saying roughly half the results were released unconfirmed.
  10. 15 October: OpenAI commits $5 million to a Simons Foundation initiative helping mathematicians examine AI results ($15 million in total with the foundation and Anthropic).
  11. 19 October: Quanta Magazine publishes the first detailed expert look at the quasi-Riemann proof.
  12. 24 October: Terence Tao's blog announces a community hub for revising OpenAI's write-ups, which it says "can be difficult to understand and can have incomplete citations".

722, 719, 372 or 377? Why the numbers disagree

Coverage uses four different counts. They're all defensible; they just count different things.

Number

What it counts

Source

722

Manuscripts (individual papers) released on 6 October

OpenAI's release post, Retraction Watch

719

Manuscripts still live after the three withdrawals on 7 October

OpenAI's GitHub repository

372

"Result families": groups of related papers, alternative proofs and papers that depend on each other

OpenAI's repository

377

"Results", as Quanta and some other outlets put it

Quanta Magazine, Ynet

The family count is the closest thing to "problems tackled", because one result can span several manuscripts. None of the pages we could read explains the 377, so treat it as another way of grouping the same work. Whatever the count, the model was set about 4,000 problems and only the successes were published.

The quasi-Riemann hypothesis in plain English

The Riemann hypothesis is arguably the most famous open problem in mathematics. It's about the zeta function, whose "zeros" (the inputs where it equals zero) control how prime numbers are spread out. Riemann showed that the interesting zeros sit inside a vertical strip between 0 and 1; the hypothesis says they all sit exactly on the line down the middle, at 1/2.

In 1896, two mathematicians showed that no zeros sit on the strip's edges, which was enough to prove the prime number theorem.

The quasi-Riemann hypothesis asks for less than the full hypothesis: just prove all the zeros live in some narrower strip. As Quanta puts it, "the narrower the strip, the more one can say about how the primes are distributed."

OpenAI released two versions:

  • The 7/8 version (dated 30 September): all the zeros lie between 1/8 and 7/8. It's 199 pages and was formally verified in Lean.
  • The 11/12 version (about a week later): a wider strip, between 1/12 and 11/12, in 49 pages. It's not yet verified in Lean. It's the only one of the released papers a human edited, and OpenAI hasn't said who.

Mathematicians mostly read the shorter one. NYU's Maksym Radziwiłł called the implications "staggering", noting that "the vast majority of consequences of RH already follow from quasi-RH".

Stanford's Jared Duker Lichtman called it "the most significant AI-for-math achievement so far, by quite a margin". Georgia Tech's Alexander Dunn called it "a surprising synthesis of well-known ideas"; it relies in part on techniques Radziwiłł and Dunn pioneered.

And the caution: Princeton's Peter Sarnak won't call it "a Theorem" until humans check every detail. Several expert write-ups exist but haven't been published.

How maths gets verified: peer review vs proof checkers like Lean

There are two ways a proof becomes trusted.

Peer review. Experts read the argument line by line, often for months, then a journal publishes it. It depends on human attention, which is why Radziwiłł's verdict that the papers "are horribly written (in a way that would lead to immediate rejection at any mathematical journal)" matters.

Formal proof checking. Lean is a programming language and proof assistant. You write the theorem and every step of the proof in code, and Lean's small trusted "kernel" checks that each step follows from the last. If it compiles, the logic holds, with no reviewer's patience required.

Lean has one big catch: it checks the theorem as written in Lean, not the English paper. If the AI translates the question wrongly, Lean will happily confirm a proof of something slightly different. On 6 October, three researchers (Bastounis, Circelli and Hansen) posted a paper arguing exactly this about OpenAI's Navier-Stokes formalisation: that the Lean proof doesn't match the natural-language argument.

So the strongest evidence is both: a Lean proof whose statement an expert has confirmed matches the real problem, plus a human-readable paper.

Which OpenAI math proofs are verified? The scorecard

Result

Status on 26 October 2026

Evidence

Erdős 1946 geometry conjecture disproved (May)

Accepted by named experts

Supporting remarks from Alon, Wood and Bloom

Ten advances (August)

Lean proofs, OpenAI says; credit disputed

Two of ten reuse uncited prior work, experts say

Quasi-Riemann, 7/8 strip

Machine-checked, not peer reviewed

Lean formalisation

Quasi-Riemann, 11/12 strip

Claimed; experts largely agree it's correct

No Lean proof yet; human write-ups unpublished

Navier-Stokes blow-up (8 September)

Claimed, disputed, under review

Lean proof contested; Clay has not ruled

300 of 719 top-line results

Lean-checked statements

OpenAI's repository changelog

Remaining ~58% of results

Claimed, unverified

English manuscripts only

14 manuscripts

Revised after errors found

OpenAI changelog, 7 October

3 manuscripts (Weil classes, Kuga-Satake, rational Hodge conjecture for K3 products)

Withdrawn

One sign error broke one paper and two that depended on it

Notice the withdrawals. One sign error in one paper took down two more that relied on it. Errors in AI output don't stay put; they spread through everything built on top.

Why mathematicians are impressed and angry at the same time

The anger isn't mostly about correctness. It's about how the work arrived.

  • Volume. Hundreds of papers landed at once, and checking is unpaid expert time. Cornell's Alex Townsend said OpenAI "should have announced the manuscripts that were lean verified first".
  • Credit. Critics say results build on human work without clear citation, as with the August results and Navier-Stokes.
  • Insight. Terence Tao argues that solving problems is only a proxy for the real goal of understanding, and warned that "indiscriminate strip-mining of open problems for solutions may destroy the ecosystem".
  • Missing data. The model's name and the thousands of failed attempts weren't released.

Others are openly excited. Harvard's Michael Douglas called it "the day when a new era of math and mathematical physics began". Both reactions can be right.

For how AI-written papers are affecting science's preprint servers more widely, see our piece on AI-generated research papers on arXiv.

The lesson for businesses: generation is cheap, verification is the bottleneck

OpenAI's model averaged about three hours per result. Checking the best one has taken the world's leading number theorists weeks, and they're not finished.

Your AI won't be proving theorems, but the shape is the same. A model drafts 200 contract summaries or a pull request in minutes; a person still has to decide which are right.

If review capacity doesn't grow with output, you haven't done more work. You have a bigger pile of unchecked drafts.

Three patterns from the maths story apply directly:

  • Survivorship. You see 372 wins, not 3,600-odd failures. If you only log the AI outputs that worked, you'll overestimate how reliable it is.
  • Cascading errors. One sign error broke three papers. One wrong figure in an AI-written report can flow into a forecast, a quote and an invoice.
  • "Verified" can verify the wrong thing. A Lean proof of a mistranslated question is worthless. A test suite that passes against the wrong spec is too.

How to build verification into AI workflows

  1. Define "correct" before you generate. Write the acceptance check first: the fields that must match the source, the totals that must reconcile, the tests that must pass.
  2. Make the machine check what it can. Lean is maths' version of this. In business it's schema validation, recomputing every number, resolving every citation and link, and running tests on generated code. Our vibe coding security checklist covers this for AI-written code.
  3. Check the question, not just the answer. Have a person confirm the AI understood the task, because a perfect answer to the wrong question still passes automated checks.
  4. Review by risk. Spot-check low-stakes output; require a named human sign-off on anything that touches money, customers or legal commitments.
  5. Label what's verified. Mark AI output as "checked" or "unchecked" so nobody builds on a draft by mistake. That was Townsend's point.
  6. Keep the failures. Log what the AI got wrong as well as what it got right, and measure accuracy over time. Our guide to evaluating a RAG chatbot shows how.
  7. Version and correct in the open. OpenAI kept earlier versions and a public changelog. Do the same for AI-generated documents, so a correction reaches everything that used the old version.

For what these results say about AI's direction more broadly, see how close are we to AGI.

Frequently asked questions

How many maths problems did OpenAI's AI actually solve?

Nobody can give a firm number yet. OpenAI released 722 manuscripts covering 372 result families from about 4,000 problems attempted, then withdrew three and revised 14. About 42% of the top-line results have machine-checked Lean proofs, and an OpenAI spokesperson said roughly half were released unconfirmed. The honest answer is "hundreds claimed, a few hundred machine-checked, a handful examined in depth by experts."

Did AI prove the Riemann hypothesis?

No. OpenAI's model claimed the quasi-Riemann hypothesis, a weaker statement that all the zeta function's zeros lie in a narrower strip, between 1/8 and 7/8 or 1/12 and 11/12. The full hypothesis, that they all lie exactly on the 1/2 line, remains open. Experts say the weaker result would still have big consequences for number theory, because many results that assume the full hypothesis follow from it.

Does a Lean proof mean an AI result is correct?

It means the logic of the formal proof is sound, which is strong evidence. It doesn't guarantee the Lean statement says the same thing as the original problem or the English paper. Researchers have argued that OpenAI's Navier-Stokes formalisation doesn't match its written argument, so experts still need to confirm the statement is faithful.

The bottom line

OpenAI's maths release is genuinely remarkable and genuinely unfinished. A model can now produce research-level proofs faster than humans can read them, and that's exactly the problem: an unchecked proof is a draft, however clever its author.

The same is true of every AI output in your business. The winners won't be the companies that generate the most; they'll be the ones that check the fastest.

Start small: write down what "correct" means for one AI-drafted output your team relies on. Want help building the checks? Book a free consultation and we'll help you design the verification steps into your AI workflow before it goes live.

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