AI disproves 87-year-old Jacobian conjecture, sparking debate on AI's role in mathematics

Extremely basic AI prompt cracks decades-old maths problem | New Scientist

A longstanding mathematical conundrum has been solved by ChatGPT in a few hours, with only a few simple prompts. The Dinitz-Garg-Goemans conjecture is a 30-year-old question in graph theory, but a counterexample posted on X by Dmitry Rybin, co-founder at AI startup Autokernel, has shown that it is false. Rybin entered just four prompts into ChatGPT 5.6 Pro: an initial one instructing the AI to "do a breakthrough and find a structured counterexample", and then three others simply urging it to continue searching. All four added up to fewer than 60 words, and the AI took a total of five and a half hours to crack the problem. Graph theory is the study of networks made up of nodes, or vertices. The Dinitz-Garg-Goemans conjecture can be thought of as a logistical challenge: imagine shipments from a warehouse to multiple locations can be split into much smaller deliveries that can be sent on different routes. The conjecture states that this scenario can be converted into another where shipments cannot be split, and that the total cost of shipping will not increase. Rybin did not respond to a request for comment, but said on X: "I know counterexamples to old conjectures are becoming a meme at this point. But I really cared about this problem and spent many weeks thinking about it." Chris Bowman-Scargill at the University of York, UK, says there's a running joke in mathematics that every conjecture in graph theory is false, just as the Dinitz-Garg-Goemans conjecture has now been proved to be. "In fields like number theory or algebra, patterns that hold for small rank [simple situations] often hold for a long time," says Bownman-Scargill. "Whereas in graph theory, structural behaviour can shift dramatically once you add just one or two vertices... which is how these conjectures continue to be made. You can see how people miss these things." AI has made rapid advances in mathematics in recent months. In May, an OpenAI model cracked a decades-old conjecture by Paul Erdős, causing a stir in mathematical circles. Earlier this week, an AI found a counterexample to the Jacobian conjecture, which had stood for nearly a century. Today, other AI users claim to have solved a Graffiti conjecture and a second graph theory problem. A website has even sprung up to catalogue the AI findings and list them by the model that was used. Abhishek Saha at Queen Mary University of London says current AI models seem particularly well-suited to problems like the Dinitz-Garg-Goemans conjecture, but there are limits to what is currently possible - and the problems solved by AI so far are of limited complexity. "AI is really good, and at least at some mathematical tasks, already superhuman," says Saha. "There is a fair bit of low-hanging fruit out there. Some conjectures can now be proved or disproved by AI with very little human input; the main challenge is simply pointing the system in the right direction. On the other hand, I don't think AI is yet at a place where it can build the theory needed to prove some of the deepest open conjectures people care about." But there are signs that AI is here to stay and will become a vital tool for mathematicians. Alexander Yong at the University of Illinois Urbana-Champaign says AI's growing role in mathematics will empower researchers to discard dead ends and push in promising directions instead. "Counterexamples to old conjectures are never quite as impressive as finding a sequence of interlocking arguments that constitute a proof," says Yong. "However, I'd expect that AI will soon prove many conjectures by a combination of their inherent superhuman energy in knowing the literature and trying many things at a prompt. Those conjectures that survive AI scrutiny will be the genuine goals for human innovation."

AI's solution to 87-year-old riddle takes mathematicians by surprise | New Scientist

A mathematician has cracked an 87-year-old conundrum with the help of AI and announced the solution unceremoniously in a tweet. The finding is the most difficult mathematical problem yet solved by AI, say experts. Levent Alpöge at Harvard University wrote on X on 19 July that the Jacobian conjecture - which academics have spent decades trying to prove was true - is actually false, giving a tiny, 216-character counterexample as proof. The Jacobian conjecture - which suggests that a certain type of mathematical function would also work in reverse - was formally set out by Ott-Heinrich Keller in 1939. It was also on an influential list of 18 fiendishly difficult problems for mathematicians to tackle in the 21st century drawn up by Stephen Smale in 1998. Alpöge did not respond to New Scientist's request for interview, but said in his tweet that part of the work was down to his "close friend fable"- seemingly referring to AI company Anthropic's Claude Fable 5. Alpöge thanked Fable for working during the World Cup final. Anthropic did not respond to a request for comment. Abhishek Saha at Queen Mary University of London says AI's recent advances in mathematics, such as the OpenAI model that recently cracked a decades-old conjecture by Paul Erdős, have been surprising, but this latest finding has stepped things up significantly. "Probably this is the biggest conjecture that AI has played a significant role [in proving or disproving] so far in mathematics," he says. "This is a pretty big deal. AI has [made] remarkable progress in the last year." The single line of mathematics posted by Alpöge was simple to verify and many mathematicians have already done so, says Saha. Now the big question is how it was done. "There are some problems that are very hard to solve but once a solution is there, they are relatively easy to check. So this is like that," he says. "I don't know how he did it, what exactly was the prompt to give Fable, because if one were to search everything, it wouldn't quite work, so obviously there was some insight also which is not currently published." Saha says both the appearance and the nature of the result is a surprise. "People have been trying to prove it [the Jacobian conjecture] because it sounds, intuitively, very true. I don't think that many people have been trying to disprove it. And now we have this one-sentence counterexample," he says. There are still open questions, says Saha. For example, this new counterexample disproves the conjecture with three variables, but a version with two variables could theoretically still be true. Chris Bowman-Scargill at the University of York, UK, says mathematicians have in some ways already adjusted to the shocking new capabilities of AI, but there is a difference between finding counterexamples that disprove conjectures and building whole new branches of mathematics, which still requires human creativity. "If you look at Fermat's last theorem [which was solved by Andrew Wiles in 1994], you had to create a hundred pages of new mathematics - you had to build a whole big theory in order to solve a conjecture," says Bowman-Scargill. "And often the interesting stuff in maths isn't 'oh, we've ticked off this conjecture, yay', it's more the stuff you have to build along the way in order to solve the conjecture." "I think [AI] has sort of proven that it can do this, so you're now like, OK, what next?," he says. Ivan Fesenko at Westlake University in China believes what comes next is increasingly capable AI models that will solve ever more complex problems, disrupting the field as they go. "Right now, AI can already produce master's degrees in mathematics. In one year, they will produce PhD degrees in mathematics. And then the question arises, do we really need so many mathematicians around if AI can do such things so nicely?" says Fesenko. "So basically we're talking about fundamental change in mathematics."

'hello there the jacobian conjecture is false thanx': why a tiny social media post has mathematicians rethinking AI

As millions of people were coming down from the excitement of the FIFA World Cup Final at the start of this week, a different kind of excitement was building within the mathematical community. Levent Alpöge, a mathematician working at the artificial intelligence (AI) company Anthropic, made a very casual announcement on X that he had found a counterexample to the Jacobian conjecture, a very old and well-known problem in a field of mathematics called algebraic geometry. He had done this using Anthropic's large language model Fable 5, released to the general public only a few weeks ago. This is just the latest of many striking mathematical breakthroughs made by mathematicians working with large language models. But this one feels a little different to those that have come before. What is the Jacobian conjecture? First, what is a conjecture? It's an idea that some mathematicians believe is true but nobody has been able to prove or disprove. Now to the Jacobian conjecture. It's fairly abstract but not too difficult to describe. The conjecture involves functions, which are like little machines which you put one or more numbers into and out pop other numbers according to some rule or equation. In this case, the functions use what are called polynomials. Specifically, it's about situations where the numbers represent points in a space, like coordinates on a map. So we can imagine that when the function takes in some numbers and puts out some other numbers, it is moving the points in space. You can test how "nicely" a function moves everything around in space by calculating something called the Jacobian determinant. If the Jacobian determinant is always a constant number that is not zero, then the function never folds or crushes space around a particular point. The Jacobian conjecture states that when the Jacobian determinant is a non-zero constant, there should always exist another function, also made up of polynomials, that reverses the original one. This will return all the points to their starting positions. Not every function is reversible. For example, if our starting function moves two of the original points onto a single point, then we cannot reverse it. Once the points have been merged, we cannot distinguish between them to send them back to the right positions. A long history of attempts - and failures The two-dimensional version of the Jacobian conjecture was stated by Czech mathematician Ludwig Kraus in 1884. It was generalised to any number of dimensions by German mathematician Ott-Heinrich Keller in 1939. It was considered so compelling that Fields Medallist Stephen Smale included it in his 1998 list of Mathematical Problems for the Next Century. During its long history, the Jacobian conjecture has been the subject of many claimed proofs, including by Beniamino Segre and Wolfgang Gröbner, two famed 20th-century mathematicians. However, in each case, subtle errors were found that invalidated the arguments. Despite this, there have also been a number of valid efforts showing the conjecture is true with various restrictions. Computational results have also shown it is true in two dimensions for polynomials up to degree 100 (that is, including powers of the variables up to 100). But nobody had proved the general case - or found an example showing the conjecture was wrong. A deceptively simple answer One of the key reasons the Jacobian conjecture is so intriguing is that, in theory, it should be easy to find a counterexample. It is straightforward to come up with examples of functions that merge points, and also examples of polynomial mappings that have a constant Jacobian determinant. However, finding a polynomial mapping with both properties is the challenge. Indeed, as one Math Stack Exchange user noted in a post from 2017, "for all what we know, some smart undergraduate can simply write a formula [...] that will be a counter-example to this conjecture". Indeed, this did turn out to be the case for Alpöge's function, which is short enough to fit into a single X post. He found an example of a function in three dimensions which has a constant Jacobian determinant of -2, and which moves multiple input points to the same output point, so it is not reversible. It shows the conjecture is false for every dimension larger than 2, with the original conjecture in two dimensions remaining open. The brevity of the counterexample made it easy for other mathematicians to verify. The latest advance in a growing series Alpöge's discovery is the latest in a string of high-profile mathematical breakthroughs made by large language models. Recent examples include OpenAI's disproof of the unit distance conjecture, and the proof of Erdős' problem 1196 by Liam Price, a 23-year-old amateur mathematician. Both examples illustrate one of the most striking strengths of AI models. They can draw on ideas from different areas of mathematics, combining them in a novel way to prove astonishing results. At the time of writing, details have not been made public regarding exactly how Alpöge prompted the AI model to produce the Jacobian conjecture counterexample and what its output looked like. However, so far this result appears to be of a different nature. Unlike many other recent AI-assisted breakthroughs, the counterexample itself is remarkably simple. The difficulty in finding it seems to have lain not in an intricate construction or a lengthy proof, but rather in finding a good way of navigating an enormous search space of possible polynomial mappings to find one with the right properties. This suggests AI may prove to be just as valuable for discovering unexpected mathematical objects as it is for constructing proofs. What this means for the future of mathematics - and human mathematicians - remains to be seen.

AI just disproved the 87-year-old Jacobian conjecture

While the world watched the World Cup final, a mathematician and an AI model quietly broke a problem that had stood since 1939. The proof fit in a single tweet. Whether it means much is the more interesting question. On Sunday evening, as Spain and Argentina played out the World Cup final, the mathematician Levent Alpöge posted a short message on X. The Jacobian conjecture, he wrote, is false. He thanked a friend for asking about it, and another friend, "fable," for working during the match. That second friend was Fable 5, Anthropic's latest AI model. Together they had produced a counterexample to a problem that had resisted mathematicians for 87 years. It was 216 characters long. What was actually broken The Jacobian conjecture was set out by Ott-Heinrich Keller in 1939. In rough terms, it says that a certain kind of polynomial map, one whose Jacobian determinant is a non-zero constant, must be reversible with a neat polynomial inverse. It became one of the field's most stubborn open problems. Stephen Smale put it on his famous 1998 list of challenges for the 21st century. New Scientist calls it the hardest maths problem an AI has yet cracked. Alpöge and Fable 5 did not prove it. They broke it. They found a single map, from three-dimensional space to itself, that has the required constant determinant yet still cannot be reversed. One valid counterexample is all it takes to sink a conjecture. The model behind it Fable 5 is Anthropic's newest frontier model. It is the public version of the system the company once called Claude Mythos, which Anthropic had described as too capable to release. Alpöge, a Harvard fellow, lists an Anthropic affiliation. The result is easy to check even if it was hard to find. The counterexample is a plain list of polynomials that any mathematician can verify by hand. Finding it was the needle-in-a-haystack part, and that is where the machine came in. It was not a one-line prompt, cautioned Bartósz Naskręcki, one of many mathematicians now waiting for Alpöge's full write-up. Searching for a counterexample like this, he said, takes real insight. Why mathematicians are stirred The reaction online was loud. Timothy Gowers, a Fields medallist, said it was the first time an AI had solved a problem outside his own area that was big enough for him to have heard of it. He stopped short of calling it the end of mathematics, since a counterexample is not a grand theory. Still, he called it amazing. He was not alone. The mathematician Daniel Litt posted at 2am that he could not stop laughing. He called it incredible. Part of the delight is historical. As the Stanford number theorist Jared Duker Lichtman noted, the conjecture is notorious for attracting wrong proofs, and it once helped sink the PhD of Yitang Zhang. He was told he had failed at it, then later became famous for work on gaps between prime numbers. An AI settling it in a tweet is, to many, poetic. It is also the latest sign of AI turning up in serious science, from biosecurity to pure maths. And rivals were circling the same target. The OpenAI researcher Aaron Lou said an internal version of the company's Codex model found essentially the same counterexample on its own. Or maybe it means little Not everyone is stirred. Andrew Blumberg, a mathematician at Columbia who sits on a project testing AI on research maths, was pointedly unmoved. "This did not cause me to update my priors," he told Mashable. This is exactly what he expects AI to be good at. His point is that a counterexample and a proof are very different prizes. A proof teaches you why something is true. A counterexample just ends the argument. Smale wanted the conjecture solved because a solution might reveal how nature is structured. This one, Blumberg argues, reveals almost nothing. "There are a lot of polynomials, and it is hard for people to check them all, but it is not hard for machines," he said. He contrasted it with OpenAI's disproof of a geometry conjecture in May, which mathematicians could unpack and build on. That is the honest shape of the story. AI is now a superb search engine for needles that humans could not sift by hand. Whether it can also explain what it finds is the harder test, and recent research suggests that question is far from settled. What comes next Alpöge hints the story may not end in destruction. Unpicking his own counterexample, he wrote that it seems to show a glimmer of a positive result hiding inside. A full write-up, he added, will follow. The work has not stopped. One developer, Alexis Gallagher, says he used GPT-5.6 to turn the single counterexample into an infinite family, one for every whole number above two. Nobody has checked it yet. And another model has already proposed a fresh conjecture to replace the one that just fell. Perhaps the eeriest footnote came from Fable itself. One user, Guy Tamam, shared what the model said when he put the proof to it. The "fable" in the credit was itself, it wrote, in a sense: the same model family, but a different conversation on a different computer, one it has no memory of. The machines broke the old rule. They are already writing the next one.

A mathematician used Fable 5 to disprove a major math problem

While the rest of the world was watching the World Cup on Sunday evening, mathematician Levent Alpöge casually announced on X that he had used Anthropic's Fable 5 to disprove the Jacobian conjecture. The Jacobian conjecture is a long-standing open problem in algebraic geometry that's bedeviled highly accomplished mathematicians for almost 90 years. It was included in "Smale's problems," a list of unresolved math problems put forward by mathematician Stephen Smale in 1998. But that proved no obstacle for Fable 5, Anthropic's latest frontier AI model. (You may remember Fable 5 as the public version of Claude Mythos Preview, the AI model that Anthropic says has such advanced cybersecurity capabilities that it was too dangerous for public release.) Alpöge is part of Harvard's Society of Fellows, and his Linkedin profile notes an affiliation with Anthropic. We've reached out to Anthropic with questions about Alpöge's work. Meanwhile, for those of us who don't understand polynomial functions from n-dimensional space, the question is: Does this represent a major breakthrough in artificial intelligence, or mathematics, or neither? "This did not cause me to update my priors about what AI can and can't do," professor and mathematician Andrew Blumberg told Mashable. "This is exactly the kind of thing I would expect AI to be able to do. If there was a counterexample that was concise and easy to state that people haven't found because it's a pain to search through all this stuff, AI will find it." Blumberg, who has a joint appointment in mathematics and computer science at Columbia University, is involved in the First Proof project, an effort to test the capabilities of frontier large language models in solving research-level mathematics. So, it's safe to say he understands mathematics and AI better than most of us. Providing a counterexample to the Jacobian conjecture is still a significant achievement for AI in mathematics, Blumberg notes. But he also says there's a big difference between providing a positive proof for a major math problem and providing a single counterexample that disproves it. Here's the metaphor Blumberg used to explain. "Suppose that Moses came down from the mountain with tablets, and on the tablet was written, 'Cancer can be cured.' Would you care? You don't just want the answer to the question. You want to learn something from the answer. And the reason Smale thought this problem was important is because he thought that if we solved it, we would understand more things about the about the way nature is structured." In other words, Blumberg says, "this counterexample tells us essentially nothing. It's just, you know, there are a lot of polynomials, and it's hard for people to check them all, but it's not hard for machines." The Jacobian conjecture isn't the only math problem AI has tackled in recent months. OpenAI announced in May an "internal model" had disproved the Erdős unit distance conjecture, a central conjecture in discrete geometry. However, in that case, the result was more productive, Blumberg says. That's because OpenAI didn't just provide a counterexample -- but a disproof of the problem itself. "The disproof there already has led to interesting things because experts in the area unpacked what was going on in the disproof and then used it to do other things," Blumberg says. And that's not necessarily the case with the Jacobian conjecture counterexample, he adds -- while conceding he's not expert enough to see if there's something special about the structure of the counterexample that we can learn from. "But it's not in and of itself interesting." Disclosure: Ziff Davis, Mashable's parent company, in April 2025 filed a lawsuit against OpenAI, alleging it infringed Ziff Davis copyrights in training and operating its AI systems.

Mathematicians grapple with a 'very rapid and very unsettling change' as AI cracks yet another century-old problem | Fortune

On Sunday afternoon, while the rest of the world's eyes were glued to the World Cup final, an AI model resolved a problem that had tortured mathematicians since 1939. By the time Kevin Buzzard woke up in London the next morning, the result had been verified. By lunch, it was all his peers at the Imperial College London's pure mathematics department could talk about; at the time of writing, Anthropic employee Levant Alpöge's post announcing the result has drawn more than 20 million views on X. "It is a big day," Buzzard told Fortune. "I think it's a great time to be alive, personally." It was the latest in a series of AI-driven mathematical breakthroughs. AI's progress (or assault) on unsolved mathematics has compounded rapidly since mid-2025, when models first solved five of six problems at the International Mathematical Olympiad. From there, the list of fallen problems grew fast: OpenAI's model disproved an 80-year-old Erdős conjecture on combinatorial geometry in May, and in June, 16 researchers from 15 universities published the Leiden Declaration on Artificial Intelligence and Mathematics, urging the profession to set guardrails around transparency, attribution, and peer review before AI reshapes what mathematical knowledge even means. Mathematicians, relegated to the role of shepherd, are left to watch AI close these questions one-by-one, reaching places where the human mind can't follow. Their reaction is a now familiar mixture of dread and amazement. An 87-year-old problem The problem is called the Jacobian conjecture and since 1939, it has rested on the work of German mathematician Ott-Heinrich Keller. Fundamentally, the problem is about what mathematicians call "maps," and the conditions under which, given a set of outputs, you can determine the input. This being math, it was based on yet another German's work a century earlier: Carl Gustav Jacob Jacobi's Jacobian determinant. The main issue for modern practitioners is that they were unable to prove Keller's conjecture true, or find a reason that it was false -- until now. On Sunday. Alpöge's result met the Jacobian determinant at every point in space -- the determinant holds steady at -2 everywhere -- yet sends three different starting points to the same destination. Meaning, it didn't pass the test. It's a "very exciting" result, Buzzard said, one that demonstrates the potential of language models to eventually reach the "supermathematician" Google Deep Learning scientist Christian Szegedy warned about half a decade ago. But it also leaves mathematicians wanting. The trouble with getting current AI models to solve pure math is that you get the "how" without the "why," explained Akhil Mathew, the University of Chicago mathematician who Alpöge credits with suggesting the problem to him. "One can check out that it's correct," Mathew told Fortune, "but it would be nice to be able to tell a story." Alpöge didn't respond to Fortune's request for comment. Why we even have pure mathematics Mathematicians have faced down automation before. Most people with high-school level math see the job as calculation, which computers conquered decades ago. "Then you go to college, and if you do some advanced math classes, you learn that actually math is all about reasoning," Buzzard said. A calculator multiplies four-digit numbers faster than any human. What a mathematician adds is the reason: told that 131 times 137 is four million, Buzzard doesn't need to reach for a calculator -- he knows two odd numbers can't make an even one. To understand something, he said, is to "make it fit in your brain," so well that you could regenerate the result yourself from the idea. The demonstration of your knowledge,in formal mathematics, is a "proof," a chain of logical steps, each following from the last, that ends at the claim you're making. Proofs are both how mathematicians build and how they're measured. A great proof can run hundreds of pages and take months of explaining to experts to trust. So far, AI doesn't yet have the capabilities to create such a proof, Buzzard said. Creating a delicate 150-page proof requires hundreds of steps, and language models have a habit of bridging gaps with plausible-sounding filler. because unlike a human colleague, the model risks no reputation by being wrong. If and when that bottleneck cracks, it'll be Buzzard's own doing, though. His career project is Lean, a popular computer language in which proofs are checked by machine rather than by exhausted PhDs. He said the proof was already checked in Lean by the time he woke up. The moment proof-writing models meet his proof-checking machine, one of human's last advantages in mathematics disappears. The question of "taste" Mathew was more tempered in his excitement, calling this moment "a very rapid and very unsettling change... especially for junior mathematicians." Michael Harris, a professor of mathematics at Columbia, wrote in a June essay in Boston Review that the AI industry treats reasoning, or understanding, as commercially worthless, and human mathematicians as a "beta version of intelligence." Yet mathematics, he argued, is one of the last examples of unalienated labor, a field that people enter, in the words of Abel Prize winner Pierre Deligne, because one can earn a living "by playing" -- what Mathew calls "telling a story." Even when Deep Blue "solved" chess by beating Garry Kasparov in 1997, people didn't stop playing chess; they learned from it. But perhaps it sounds uninspiring for the public to subsidize mathematicians to play. Even before AI threatened their work, federal funding for mathematics research has fallen roughly 72% under the Trump administration's cuts to the National Science Foundation. PhD admissions at top research universities are down 15% this fall, the second consecutive year of contraction; George Washington University's math doctorate will admit no funded students at all. Some think that the death of mathematical professionalization is good, that the machines democratize the whole enterprise, the "playing." Garry Tan, president of Y Combinator, reacted to the news on X by hailing the return of the age of the "gentleman scientist" -- rich savants à la Benjamin Franklin, funding their own curiosity. But Alpöge is no hobbyist; he is a Harvard valedictorian who has spent a decade using algorithms to calculate exactly this kind of problem. And that might be the trick to keeping humans in the mathematical loop, Buzzard said. Beyond calculating, beyond even logical reasoning, "understanding," at the bottom, is knowing what to ask, what Silicon Valley has taken to calling "taste." "People have tried to get machines to ask questions, and they're abysmal," he said. "All the questions they ask are either boring or obviously true or obviously false." The monuments of the field -- the Riemann hypothesis or Keller's Jacobian -- are named for the people who posed them, not the people who settled them, Buzzard pointed out. "It's not a coincidence. You have to be a brilliant mathematician to come up with the right question."

A Harvard mathematician and an AI model may have cracked an 87-year-old math problem

A young mathematician teamed up with a new AI model to tackle one of the hardest open problems in mathematics, and the result is causing a stir among mathematicians and AI researchers on X. Levent Alpöge, a 33-year-old researcher at Harvard University, spent Sunday working with Anthropic's Fable model to defeat the Jacobian Conjecture, a notoriously difficult problem proposed by German mathematician Eduard Ott-Heinrich Keller in 1939. The challenge for Alpöge and Fable was to find a counterexample proving the conjecture false (a single verified counterexample would be enough). Alpöge casually announced the result on X, thanking both the friend who encouraged him to take on the problem and Anthropic's Claude Fable, the AI model that helped him do it. The conjecture proposes that if you start with a point on a grid represented by ordinary variables (say, x and y), then use polynomial equations -- equations involving powers such as x² and y³ -- to create new coordinates, you should be able to reverse the process using polynomial equations and recover the original coordinates. (For a deeper explanation, visit the Wikipedia page.)