How Far Has AI Gotten on Unsolved Math Problems? What 120 Records Show About the Last Two Years

A “solution” to a Millennium Prize Problem has been announced

On September 8, 2026, OpenAI announced that an AI it is developing in-house had produced a proof resolving a problem about the Navier–Stokes equations. It is one of the Millennium Prize Problems: seven hard problems chosen in 2000 by the Clay Mathematics Institute in the United States, each carrying a prize of one million dollars. The only one recognized as solved so far is the Poincaré conjecture, solved in 2002–2003 by the human mathematician Grigori Perelman.

The Navier–Stokes equations describe how water and air flow, and they are used every day in weather forecasting and aircraft design. Yet no one had proved that the equations always give a meaningful answer, that is, that the speed of the flow can never become infinite at a single point.

What OpenAI showed is an example in which the speed does become infinite, under the condition that a force keeps being applied to the flow from outside. It ran about 10,000 AI agents for 88 hours and wrote a 166-page proof. The Clay Mathematics Institute said the problem “appears to have been solved,” while stating that it will not rush its review.

Mathematicians believe the proof is correct because it passed a machine check. At the same time, some said it is “not written for humans” and that “it is very hard to extract understanding from this proof.” A dispute over credit also continues with a researcher who had published results on the same problem the day before.

This one case contains nearly every issue of the past two years in AI and mathematics: a big announcement, results still being checked, machine verification, and disputes over credit.

This article collects announcements from 2023 through September 2026 in which AI was said to have contributed to an unsolved math problem, checks each one against papers and official records, and uses the resulting 120 cases to lay out what has happened.

Where do the widely reported cases stand now?

• October 2025　“GPT-5 solved 10 Erdős problems” (OpenAI)Already known
  What it showed: The answers to the 10 problems were already in published papers. What the AI found were those papers.
  Current status: The announcement post was deleted.

• May 2026　Disproof of the unit distance conjecture (OpenAI)Confirmed
  What it showed: A conjecture about distances between points in the plane, posed by Erdős in 1946, is false.
  Current status: Nine outside mathematicians wrote a verification paper.

• July 2026　Counterexample to the Jacobian conjecture (researchers at Anthropic)Confirmed
  What it showed: A conjecture about polynomials, open since 1939, is false for three or more variables.
  Current status: An expository paper has been published, and there is a machine check.

• August 2026　Zeros of the zeta function (Anthropic)Confirmed
  What it showed: More than two thirds of the zeros of the zeta function, which governs how prime numbers are distributed, lie on the line where they are expected to lie. This is not the Riemann hypothesis itself.
  Current status: There is a machine check, and human researchers have also published a separate proof.

• September 2026　Lean formalization of Fermat’s Last Theorem (Anthropic)Confirmed
  What it showed: A theorem proved in 1995 was rewritten in 11 days into a form a machine can check (13 million lines). This is not new mathematics.
  Current status: Kevin Buzzard, a leading figure in formalization, has confirmed it.

• September 2026　Navier–Stokes equations (OpenAI)Not yet verified
  What it showed: See the opening section.
  Current status: The Clay Mathematics Institute is reviewing it.

• September 2026　“Solved over 100 open problems” (OpenAI)Not yet verified
  What it showed: Announced as results of a new in-house model.
  Current status: As of September 27, neither the list of problems nor the proofs had been published.

The conjectures that make the news are ones that have been known for decades. As the chart shows, even conjectures from the 1930s have moved. However, results that have been fully checked (green) are mixed with results still being checked (gray). “Shown false” in the chart means the AI found an example proving the conjecture wrong. Showing that a conjecture is false also counts as solving the problem.

How has the number of announcements changed over two years?

From summer 2025, the number of announcements rose sharply. There were 3 announcements in April–June 2025, 20 in October–December, and 35 in July–September 2026. Those 35 in the latest three months equal 40% of the total for the preceding two and a half years (85).

It was not only the number that changed. The kind of AI used changed too. Among the 13 cases through June 2025, only 3 had a large language model such as GPT or Gemini work on the problem directly; most used special-purpose programs built for mathematical search. Among the 107 cases from July 2025 onward, 81, about three quarters, used large language models such as GPT, Gemini, or Claude (an approximate count based on the names of the AI systems used).

Over the same period, AI reached the level of human gold medalists at the International Mathematical Olympiad for high school students: a silver-medal-level score in 2024, gold-medal level in 2025, and reports of perfect scores in 2026. From 2026, research organizations that evaluate AI began using real unsolved problems themselves as tests.

However, of the 120 cases announced as “solved,” only 64, about half, have been confirmed by third parties. Of the rest, 40 have not yet been verified by anyone, 11 already had answers in published papers, 4 had wrong proofs, and 1 drew criticism for insufficient citation of prior work.

In this article, a result is called “confirmed” if it meets at least one of the following three conditions.

• It passed expert peer review and was published in a journal or similar venue
• It was verified in a language for machine-checking proofs, such as Lean, and the record is public
• At least two experts other than the authors publicly stated that it is correct

Who is leading the work?

We sorted the 120 cases into four groups by who mainly drove the work (the method is in Appendix 1).

• In-house teams at major AI companies (55 cases): OpenAI, Google DeepMind, Anthropic, and Meta. Almost all of the cases that get major press coverage come from here.
• AI-for-math startups (5 cases): Harmonic, Axiom Math, Math Inc, and others. Few in number, they handle rewriting known proofs for machine checking and building tools for machine verification.
• Researchers at universities and institutes (31 cases): Researchers using AI in their own work. The quantum computing theorist Scott Aaronson wrote that GPT-5 supplied the key step in a proof he had been stuck on.
• Individuals and online volunteers (29 cases): Mainly people posting on the Erdős problems website. Most of the 4 wrong proofs and the cases that were already known also fall here.

Major companies make the headlines, but by count, more than half of the cases come from researchers and volunteers. And professional mathematicians in the relevant fields are deeply involved even in the major companies’ big results, as a later section shows.

“Solved” does not mean just one thing

Of the 120 cases, 43, just over a third, claimed to have solved a problem completely. The rest include improving a record for an upper or lower bound (22 cases), solving special cases (21 cases), and finding a counterexample to a conjecture (14 cases). A headline saying “AI solved it” can mean any of these.

Of the 120 cases, 43 concern “Erdős problems”: more than 1,100 problems that the 20th-century mathematician Paul Erdős left in his papers and talks over his lifetime, now collected on a website by the mathematician Thomas Bloom. All 4 wrong proofs were Erdős problems, and so were 9 of the 11 cases that were already known.

There are two reasons for this.

The first is the nature of the problems. “Open” on the website means the maintainers do not know the answer. So the list mixes problems that top mathematicians could not solve in decades with problems that nobody ever seriously looked at, or that had in fact been solved somewhere in the literature. The records kept by Terence Tao, the mathematician who received the Fields Medal in 2006, and his collaborators also warn that “open for years” can mean “nobody looked seriously” rather than “hard.” The October 2025 “10 problems solved” episode was exactly this. Even then, the AI did find many old papers and papers in other languages that humans had overlooked. The problem was claiming it had “solved” them. That said, a problem nobody had worked on is not necessarily an unimportant one. A new field can grow out of a problem no one was paying attention to (Euler’s bridge problem, discussed later, is one example).

The second is a bias in the records. Only for Erdős problems do Tao and his collaborators publicly log each case, including not just successes but also errors and cases that were already known. Other fields have no such ledger. So this chart shows not only that “Erdős problems have many stumbles” but also that “Erdős problems are the only place where stumbles are recorded.” Failures in other fields should be assumed to surface less often.

Who checks correctness, and how?

AI produces plausible-looking mistakes, so checking is essential. Who does the checking has changed over these two years. Of the 64 confirmed cases, 57 were proofs checked by machine.

The tools are languages such as Lean. When a proof is written step by step in a form a machine can read, the machine determines whether there are any gaps or errors in the logic. The rewriting of Fermat’s Last Theorem mentioned above works the same way.

The 5 cases confirmed through expert peer review were all announced in 2023–2024. Peer review in mathematics usually takes one to two years, so announcements from 2025 onward have probably not yet received review results.

Machine checking has limits. What Lean guarantees is only that “the statement as written is true.” If a mistake is made when translating the original problem into the machine’s language, a correct proof still solves a different problem. A Google DeepMind paper reports cases in which the AI left hard steps marked “to be proved later” and cases in which it invoked theorems that do not exist. Checking that the translation is right is still a human job.

Human checking is not perfect either. In a paper in which researchers audited OpenAI’s 10 announced results by hand, one of the points flagged as an “error” turned out to be a bar over a symbol that had simply disappeared when the text was extracted from a PDF.

Why is so much computing power used?

For Navier–Stokes, about 10,000 AI agents ran for 88 hours. Why does it take that much computation?

Because even when a mathematical answer is short, finding it takes a long time. The counterexample to the Jacobian conjecture is a formula made of three polynomials. Checking it takes a moment, but finding it required trying countless candidates. It is like a PIN code: checking the right answer is instant, but finding it means trying many combinations.

Mathematicians, too, have spent years trying promising paths one after another and turning back at dead ends. AI runs that trial and error in parallel, at enormous scale. It is less a brute-force search over numbers than a brute-force search over proof strategies.

Proofs supported by computer calculation existed before AI. The four color theorem (any map can be colored with four colors), proved in 1976, used a computer to check its cases.

Has AI created new ways of thinking, as great mathematicians have?

The history of mathematics includes people who, while working on a single problem, changed how the problem itself is seen.

In 1736, the mathematician Leonhard Euler answered the problem of the “Seven Bridges of Königsberg.” Seven bridges connected an island in the river and the two banks of a town, and the question was whether one could walk across all seven bridges exactly once each. The townspeople tried many routes, but no one managed it.

Euler stopped trying routes. He set aside the shape of the land and the length of the bridges and looked only at how many bridges touch each landmass. On any landmass you pass through along the way, every bridge you enter by is paired with a bridge you leave by, so the number of bridges there must be even. Only two landmasses, where you start and where you finish, may have an odd number. In this town, however, all four landmasses had an odd number of bridges (5, 3, 3, and 3). So the walk is impossible in any order. This way of “thinking only in dots and lines” later became the field called graph theory, which leads to today’s route-finding in car navigation systems and much more.

In the 19th century, Évariste Galois, working on whether fifth-degree equations have a formula for their solutions, stopped searching for the answers directly and instead studied the ways the answers can be swapped with one another. The idea that came out of this, the “group,” is now a foundation of mathematics and physics.

What about AI? Reading the 120 cases, AI’s work can be roughly divided into the following three stages (this classification is our own).

1. Searching: Trying huge numbers of candidates to improve records or to find examples that break conjectures. Most of the record improvements (22 cases) and counterexamples (14 cases) take this form. The counterexample to the Jacobian conjecture belongs here.
2. Connecting: Bringing a tool that was well known in one field to a problem where no one had thought to use it. Erdős problem #1196 is an example.
3. Creating: Building a new way of thinking itself, which becomes a new field, as with Euler’s graph theory or Galois’s groups. Whether any AI result falls into this category cannot be judged for decades.

In Erdős problem #1196, the AI brought ideas from probability to this problem about whole numbers. This approach is said to have been overlooked by experts ever since Erdős’s 1935 paper. It resembles, in some ways, how Euler changed the way a problem was seen. Of this proof, Tao said that “a new way of studying the properties of large numbers has been found.” At the same time, Tao said it is “not yet clear” how important this method will turn out to be (Scientific American).

The Navier–Stokes result from the opening section is, by the size of the problem tackled, the largest of the results AI has been involved in. But what was solved is the case in which an outside force keeps being applied. Mathematicians have said that “there is little to learn from this proof.”

Whether new fields like graph theory or group theory will grow out of AI’s results is not yet known. Galois’s ideas themselves took more than a decade to be understood by mathematicians.

On the other hand, it is not accurate to dismiss AI’s work as “mere brute force.” Choosing which directions are worth searching and what to connect with what involves both human mathematicians’ judgment and the AI’s suggestions. There are cases in which the AI chose the direction itself (some Erdős problems, and the counterexample to the Köthe conjecture), and cases in which mathematicians set the direction and left the rest to AI (the zeros of the zeta function).

Who notices the value of a result?

In April 2026, 23-year-old Liam Price entered Erdős problem #1196 into GPT-5.4 Pro and got a proof. What follows is based on an article in Scientific American (April 24, 2026).

Price says he has no advanced mathematical training and did not know what the problem was. He sent the result to Kevin Barreto, an undergraduate studying mathematics at the University of Cambridge with whom he sometimes works on Erdős problems. Barreto recognized its importance and alerted experts.

The problem had been open for about 60 years. Jared Duker Lichtman, a mathematician at Stanford University, had solved a problem of the same kind in his doctoral thesis but had been stuck on #1196. Tao explained that everyone who had attacked it before had all taken a slightly wrong turn at the very first step. The AI used a tool that was well known in a neighboring field but that no one had thought to apply to this problem.

However, according to Lichtman, the AI’s original write-up of the proof was “pretty bad.” Experts had to sift out what it was trying to say. Lichtman and Tao rewrote the proof more concisely, found that the same method works on other problems, and wrote a paper with Price and others as coauthors.

The same structure appears in the big results of major AI companies. Four of Anthropic’s major announcements (the counterexample to the Jacobian conjecture, the zeros of the zeta function, the construction of special matrices, and a new record for elliptic curves) all involved the same mathematician, Levent Alpöge. It is professionals in the field who decide which problems to work on and whether the results are valuable.

What AI has made dramatically cheaper is the “solving” part. Choosing what to solve, noticing the value of what comes out, and extending it into further research remain, for now, on the side of humans with mathematical ability.

This shift is also visible in where money goes. The computing cost of having AI solve a problem varies greatly from problem to problem. Google DeepMind reported that when AlphaProof Nexus solved Erdős problems and others, the computing cost was a few hundred dollars per problem. For Navier–Stokes, on the other hand, about 10,000 AI agents ran for 88 hours. Even so, compared with work that takes experts years, the cost of “solving” has dropped sharply. Meanwhile, startups promising technology that guarantees the correctness of proofs by machine have been valued at over a billion dollars, such as Harmonic at 1.45 billion dollars (November 2025). Major AI companies have begun hiring mathematicians as researchers and giving them the job of choosing problems and checking results.

So where do people with the ability to choose what to solve and notice the value of results come from? One clue is the International Mathematical Olympiad for high school students. The economists Ruchir Agarwal and Patrick Gaule followed Olympiad participants over their later lives. They found that those who scored higher as teenagers went on to write more papers as adults and were more likely to win the Fields Medal.

On the other hand, many winners of major mathematics prizes never competed in the Olympiad. Many of them belong to generations before the competition began, or come from countries that did not take part. They had ability but no venue in which it could be measured and noticed. Agarwal and Gaule’s research also showed that among participants with the same Olympiad score, those from low-income countries later produced 34% fewer papers and received 56% fewer citations than those from high-income countries. The paper’s title is “Invisible Geniuses.” Talent turns into results only when there is a place that finds and develops it.

Why are leading mathematicians speaking out?

In 2026, mathematicians’ voices became more prominent. In June, the “Leiden Declaration,” which urges people not to believe exaggerated claims about AI’s mathematical abilities, was issued and endorsed by the International Mathematical Union. On September 8, Tao issued a warning, and on September 11, 25 Fields Medalists released a statement.

Among those speaking out are the mathematicians who have used AI most actively. Tao has collaborated with AI on Erdős problems and publicly recorded where AI contributed. He has also reported that an inequality in one of his own papers was solved with AI’s help. These voices are not saying “don’t use AI.”

The concerns that can be read from the statements and remarks fall into four groups.

1. Speed: Tao said AI companies are solving problems “faster than the mathematical community can digest.” The concern is that new results keep arriving and there is no time to understand them, write them into textbooks, and teach them to students.
2. How credit is described: Who did what, and was prior work cited properly? Mathematicians pointed out that two of OpenAI’s results did not sufficiently cite earlier work.
3. Announcing before checking: Press releases and social media posts keep coming before proofs are published or checked.
4. The purpose of research: The Fields Medalists said that the purpose of mathematics is not just to solve problems but to deepen human understanding, and pointed to a “severe misalignment” with the trend of treating problems as material for comparing AI performance.

Some ways of announcing results have been praised. The British mathematician James Maynard acknowledged that AI made a genuinely interesting contribution in Anthropic’s zeta function announcement, and praised the announcement for being restrained and for giving credit to prior work.

Constructive proposals have also appeared. Daniel Litt of the University of Toronto wrote that the purpose of mathematics is human understanding and that doctoral examinations should put more weight on whether candidates can explain their work orally. In September, OpenAI also set up an advisory group of nine outside mathematicians, including Tim Gowers.

The mathematicians’ statements can be read as the mathematical community itself rethinking, now that AI can produce proofs cheaply, how to define researchers’ work, how to evaluate it, and how to train young researchers.

What do humans do when working with AI?

From here on, this is our view, based on reading these records.

In their statement, the 25 Fields Medalists said that solving problems is only part of the purpose of mathematical research, and that this concern extends to intellectual work beyond mathematics. They did not, however, propose specific rules for what to do.

Putting the 120 cases side by side with the examples of Euler and Galois, the work humans do when working with AI seems to fall into the following five parts.

1. Choosing the problem: In four of Anthropic’s major results, the mathematician Alpöge decided which problems to work on. AI is also reaching problems that nobody had seriously looked at, such as the Erdős problems. But as the Fields Medalists pointed out, if problems come to be chosen as material for comparing AI performance, AI computing and researchers’ attention may end up concentrated on famous problems alone. That is why it matters that humans, not company interests or popularity, decide what to solve.
2. Knowing where earlier attempts got stuck: For #1196, Tao explained that the mathematicians who had tried it before all took the same path at the first step and got stuck further along. Knowing where others got stuck gives a clue for trying a different path.
3. Deciding how to attack: Euler decided to stop trying routes one by one and to count only the number of bridges touching each landmass. In #1196, the AI took on this part.
4. Checking that it is real: There are two things to check. The first is whether the answer is already known. The problems that GPT-5 was announced to have “solved” in October 2025 already had answers in published papers. The second is whether the proof is correct. The proof of #1196 was rewritten in Lean, a language for machine-checking proofs, and passed the machine check. But what Lean checks is only that the proof correctly solves the problem as rewritten. Whether the rewritten problem is the same as the original must be checked by people.
5. Understanding and extending it: According to Lichtman, the AI’s original write-up of the proof was “pretty bad.” Lichtman, Tao, and others worked out what it was trying to say, rewrote the proof more concisely, and showed that the same method also solves other conjectures. It was at this stage that the result became valuable as “a new way to solve a 60-year-old problem.”

We believe these five parts apply outside mathematics as well. When developing a new business or a patentable idea, simply handing the task to AI tends to return ordinary ideas. People look into earlier cases, have the AI propose candidate approaches borrowed from other industries, and decide which one to pursue. What differs from mathematics is how the fourth part, checking, works. In mathematics, a machine can check that a proof is correct; whether a business idea is good is decided by the people who use it and by the market.

Summary

• Announcements that “AI solved an unsolved math problem” rose sharply from summer 2025, with 35 in the last three months alone.
• However, only about half have been confirmed by third parties. When you see such a headline, checking three things tells you what it really means: “Is it the actual problem, or a related one?” “Who checked it?” and “Was the answer already known?”
• Checking correctness has increasingly been taken over by machines. But whether the problem has been correctly translated into the machine’s language is still checked by people.
• Most of AI’s work is “searching,” trying huge numbers of candidates, and “connecting,” bringing in tools from other fields. Whether new fields will grow out of it is not yet known.
• What AI has made cheaper is the “solving” part. The role of mathematicians with the ability to choose what to solve, notice the value of results, and extend them has, if anything, become clearer.

Most AI companies publish only their results. The process, where humans set the direction, what the AI produced, and where it went wrong, is almost never made public. That is exactly what mathematicians are asking for.

Roppongi Venture Capital runs “Agora,” where young talents discuss ideas with AI and people from other fields, create things that do not yet exist, and publish their thinking process as it happened (record of Session 1).