OpenAI cracks 80-year-old Erdős problem, stunning mathematicians with AI's biggest math breakthrough

OpenAI claims it solved an 80-year-old math problem -- for real this time | TechCrunch

OpenAI claims its new reasoning model has produced an original mathematical proof disproving a famous unsolved conjecture in geometry, which was first posed by Paul Erdős in 1946. If this sounds familiar to you, it's because this isn't the first time OpenAI has made such a bold claim. Seven months ago, the AI giant's former VP Kevil Weil posted on X: "GPT-5 found solutions to 10 (!) previously unsolved Erdős problems and made progress on 11 others." It turns out, GPT-5 didn't actually solve those problems; it just found existing solutions that already existed in the literature. Taunts from rivals like Yann LeCun and Google DeepMind CEO Demis Hassabis followed, and Weil promptly took down his premature post. Today, at least, it seems OpenAI didn't make the same mistake twice. Alongside the announcement, OpenAI published companion remarks in support of the disproof from mathematicians like Noga Alon, Melanie Wood, and Thomas Bloom, who maintains the Erdos Problems website, and previously called Weil's post "a dramatic misrepresentation." "For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids," OpenAI posted on X. "An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better." The company said this marks "the first time AI has autonomously solved a prominent open problem central to a field of mathematics." The proof, per OpenAI, came from a new general-purpose reasoning model, not a system specifically designed to solve math problems or even this problem in particular. OpenAI says this is significant because it means AI systems are now more capable of holding together long, difficult chains of reasoning and connecting ideas across fields in ways researchers may not have previously explored. That has implications for biology, physics, engineering, and medicine. "AI is helping us to more fully explore the cathedral of mathematics we have built over the centuries," Bloom said in a statement. "What other unseen wonders are waiting in the wings?"

AI just solved an 80-year-old 'Erdős problem,' and mathematicians are amazed

I agree my information will be processed in accordance with the Scientific American and Springer Nature Limited Privacy Policy. We leverage third party services to both verify and deliver email. By providing your email address, you also consent to having the email address shared with third parties for those purposes. After 80 years of fruitless struggle by human mathematicians, a major geometry conjecture has at last been solved -- via a straightforward query to a chatbot. The company OpenAI, maker of ChatGPT, announced the result yesterday, together with comments from a number of stunned experts who declare the AI's method "clever" and "elegant." The achievement follows months of loudly reported but less impressive AI-powered advances in mathematics, and marks a true milestone. Unlike all those previous feats, this result would merit publication in a top math journal as well as major media attention, even if it were performed by humans alone. "No previous AI-generated proof has come close" to meeting those high standards, wrote Tim Gowers, a mathematician at the University of Cambridge, in commentary solicited by OpenAI. If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. "This is the unique interesting result produced autonomously by AI so far," says Daniel Litt, a mathematician at the University of Toronto with no involvement at OpenAI. The "unit distance" problem is simple to explain but formidable to solve -- a mathematician's favorite quality. Draw nine dots on a sheet of paper. The goal is to get as many pairs of dots as possible to be an inch apart. You can put them all in a line, so that you have eight pairs separated by an inch. Or you can draw a three-by-three grid, and count 12 pairs. For any number of dots, even billions or trillions, the problem asks: what's the highest number of pairs you can get? In 1946, the mathematician Paul Erdős made a guess at the best strategy. It's the grid approach, but with a much smaller spacing between dots, so pairs can be established across several grid points. Erdős showed that by using sophisticated mathematics to choose this spacing extremely carefully, you can do slightly better than a simple grid. But only slightly. In fact, Erdős claimed that no one could do better. And despite valiant efforts, for eight decades no one did. But no one managed to prove him right either, even though most experts agreed with his intuition. That changed two weeks ago, when OpenAI mathematicians Mehtaab Sawhney and Mark Sellke -- who have made headlines recently for using AI to solve a number of less prestigious "Erdős problems" -- fed the conjecture to an internal large language model (LLM) trained for general reasoning. They asked it whether Erdős was right. After churning out hundreds of pages of careful logic and calculations, it beat his long-standing record. "It feels like magic," Sawhney says. "It's kind of an amazing experience to have a machine give back something which really resembles how I work." "What the model did is totally different from the 'square grid' construction," Sellke says. It instead constructed a more elaborate grid, one living in a kind of higher dimension. This higher-dimensional lattice of points has special mathematical symmetries that facilitate separating even more pairs by the same distance. The AI model then developed a way to map this otherworldly grid back down to the two-dimensional page, producing a flattened numerical "shadow." The result is far from a grid, and Sawhney says it's too difficult to actually draw on paper, even for a small number of dots. The AI did not prove that its approach is the best anyone can do, though. In fact, the mathematician Will Sawin has already improved upon the AI's grid, although he hasn't posted his proof yet. OpenAI privately contacted Litt, Sawin and a number of other mathematicians to verify the LLM's proof. Together (and without the company's direct involvement), they then wrote up their individual takeaways. (No external experts have seen the AI's original output, however -- just an edited version of its train of thought.) What stood out, they said, was the AI's preternatural patience and focus. Human experts, largely agreeing with Erdős's thinking, had spent more effort over the years trying to prove rather than disprove the conjecture. And even those few who looked for a counterexample would be unlikely to follow such a difficult and tedious path -- constructing this high-dimensional shape -- without any enticing hint of success. But an LLM experiences the costs and benefits of trial and error differently. "AIs have an edge: It's not just that they can try all known methods," says Jacob Tsimerman, a mathematician at the University of Toronto who was not involved in the work. "They can play for longer and in more treacherous waters than mathematicians without getting overwhelmed." Several of the experts consulted by OpenAI noted that while the unit distance problem was well-known, a proof that Erdős was right would have been far more mathematically rich than a counterexample. Such proofs usually necessitate totally new insights that can then be applied to a wider range of problems. The mathematical tools the AI used here are not novel, although their application in this domain appears to be. "The model did not invent something fundamentally new that nobody saw coming," says Sébastien Bubeck, a mathematician leading OpenAI's mathematical explorations. "It just executed like an amazing mathematician." The experts also hastened to add that, without humans intervening to "clean up" the AI's work, the result wouldn't be so convincing. "The human still plays a vital role in discussing, digesting, and improving this proof, and exploring its consequences," wrote mathematician Thomas Bloom in the "reflections" document. Harvard mathematician Melanie Matchett Wood writes in her commentary that if the assembled human experts had combined their efforts for the same amount of time it took them to simply parse the LLM's answer, "the mathematicians would have found a counterexample." This is plausible because the AI's solution was a straightforward (in hindsight) approach that no human had ever attempted despite the tools already existing. Such circumstances are thought to be uncommon for major unsolved math problems. "I guess it got lucky that it found one of the cases where experts tried and missed something," Litt says. Genuinely new, groundbreaking ideas remain beyond the reach of current LLMs, instead leaving the machines to mine the literature for rare gems where humans missed a relatively simple approach. Even so, Litt adds, "my guess is we're about to find out they're actually not that rare." In her commentary, Wood also warned of AI's less-desirable traits as a mathematician, like its tendency to present every idea as its own. "Our professional norms require us to cite previous work whose ideas influenced our work," she wrote. "ChatGPT is in some sense 'familiar' with all the previous work." What this will do to mathematics -- a field currently populated by humans driven by a thirst for knowledge and a passion for the elegant beauty of mathematical truth -- is the bigger question, Wood concluded. "We urgently need to plan for how we can keep our work rigorous and correct."

Mathematicians stunned by AI's biggest breakthrough in mathematics yet

Artificial intelligence built by OpenAI has cracked a decades-old conjecture by Paul Erdős, which mathematicians have hailed as a monumental moment for AI in mathematics An 80-year-old maths conjecture that has eluded the world's greatest mathematicians has been cracked by an artificial intelligence model built by OpenAI. The result has stunned experts and is being hailed as a seismic moment for AI's mathematical ability. "This is a problem that I didn't expect to see solved in my lifetime," says Misha Rudnev at the University of Bristol, UK. "It's absolutely a bomb." Tim Gowers at the University of Cambridge wrote that the solution is "a milestone in AI mathematics" in a blog post accompanying the work. "If a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that." Twentieth-century mathematician Paul Erdős considered the puzzle, known as the planar unit distance problem, as his "most striking contribution to geometry", because it was seemingly simple to explain but deeply complex to answer. He asked: if you take an infinite-sized piece of paper and draw a number of dots in a pattern of your choice, what is the maximum number of equal-sized lines you can draw between these dots? Erdős conjectured that the patterns that yielded the most connections were points arranged in a grid, meaning the maximum number of connections would be only slightly higher than the number of points themselves. Successive attempts to prove that this really is the upper limit, or find a different arrangement of points that might lead to many more connections, yielded only small successes. The most recent improvement to Erdős's conjecture was more than 40 years ago. Now, a model from OpenAI has found that Erdős was significantly wrong, and that you can arrange points in less symmetric patterns that can yield a far greater number of pairs. "My immediate reaction was disbelief," says Will Sawin at Princeton University. "I thought the way that it was trying to solve it wouldn't work, but then I looked at it more and I convinced myself that it does work. I pretty quickly became convinced this is the most significant achievement by AI in mathematics so far." OpenAI hasn't said exactly how the model differs from publicly available AIs or how it was trained, but the firm's researchers have publicly commented that the model is "general purpose" and wasn't trained "with the goal of doing math research". The AI borrowed a technique from algebraic number theory to construct vast lattices in much higher dimensions than the two of a plane. Once it had identified and built these more complex shapes, it then collapsed them down to two dimensions, producing a shadow of the higher-dimensional shapes. "The counterexample discovered by the AI is complex, and although the ideas to produce it were already in the literature, it certainly takes some ingenuity to put them together," says Kevin Buzzard at Imperial College London. While the result is impressive, it is also partly a consequence of the fact that mathematicians didn't even consider that Erdős's original conjecture may have been false, says Samuel Mansfield at the University of Manchester. UK. Even if mathematicians did experiment with disproving it, very few geometry specialists would have then been knowledgeable enough in advanced number theory to do so. "This is something that requires you to know a lot about multiple areas," he says. "In retrospect, it's maybe not so surprising. This seems to be what an AI would absolutely be good at doing." The main appeal of the problem was the "pure intellectual challenge", says Rudnev, and it may not have any particular ramifications for other outstanding problems, but it has already sparked some further work. After seeing the proof, Sawin used the technique that the AI had discovered to produce a slightly improved higher number of how many points could be joined together. "Like many other AI breakthroughs, it did not take humans long at all to internalise, understand and generalise the arguments," says Buzzard. "One can contrast this with some human breakthroughs which have taken the community months or years to validate."

80-year-old geometry puzzle cracked by OpenAI using number theory

For nearly 80 years, mathematicians believed they understood the limits of a famous geometry puzzle first posed by legendary Hungarian mathematician Paul Erdős. Now, an AI model developed by OpenAI has overturned that assumption and solved one of the field's most stubborn open problems. The breakthrough centers on the "unit distance problem," a deceptively simple question that asks how many pairs of points can sit exactly one unit apart on a flat plane. Despite its simplicity, the problem has challenged mathematicians since 1946 and became one of the best-known questions in combinatorial geometry.

OpenAI makes breakthrough on 80-year-old maths problem

Company says work on Paul Erdős planar unit distance problem shows advance in AI reasoning OpenAI has claimed a further advance in AI reasoning after its technology successfully tackled an 80-year-old maths problem. The company behind ChatGPT said it had made a breakthrough with a challenge first posed by Hungarian mathematician Paul Erdős in 1946: the planar unit distance problem. The question posed by Erdős is simple to explain. If you take a sheet of paper and add some dots, how many pairs can be the same distance apart? Erdős proposed the number would rise only slightly faster than the number of dots themselves. OpenAI's model concluded otherwise by drawing on different branches of mathematics to uncover a family of arrangements that break the limit in Erdős's conjecture. "For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids," OpenAI wrote on X. "An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better." While the work has excited mathematicians, the broader problem remains unsolved because the AI did not come up with a new answer for how fast the pairs of dots rise, but merely showed that the limit Erdős proposed was too low. OpenAI, which is preparing to float on the US stock market, said the calculations had been made by a general-purpose reasoning model - which breaks down problems into smaller steps - rather than a system trained specifically for mathematics. The startup has been tripped up before by its attempts to solve Erdős's problems, having hailed a supposed breakthrough last year that was in fact based on already existing literature absorbed by the model. This time, OpenAI's work has been validated by mathematicians, including Thomas Bloom, a mathematician who maintains the Erdős problems website and criticised OpenAI's prior Erdős claims. Bloom co-authored a companion paper to OpenAI's blog post flagging the Erdős achievement. Bloom wrote that the AI system had attained its results by "persevering down paths that a human may have dismissed as not worth their time to explore". However, he added that humans had been involved in the AI's work. "While the original proof produced by AI was completely valid, it was significantly improved by the human researchers at OpenAI and the many other mathematicians involved in the present paper. The human still plays a vital role in discussing, digesting and improving this proof, and exploring its consequences," he wrote. Mathematician Tim Gowers, also writing in the companion paper, described the result as "a milestone in AI mathematics". Andrew Rogoyski, of the Institute for People-Centred AI at the University of Surrey, said the announcement showed AIs were giving humans new ways to look at problems. "It's becoming clear that AI is impacting the world of creative thought and will become a fundamental tool of future scientific research," he said.

OpenAI reasoning model disproves 80-year-old Erdős geometry conjecture

OpenAI claims its new reasoning model has produced an original mathematical proof disproving a famous unsolved conjecture in geometry, first posed by Paul Erdős in 1946. This announcement follows a previous assertion by OpenAI's former VP Kevin Weil, who seven months ago claimed that GPT-5 found solutions to 10 previously unsolved Erdős problems and made progress on 11 others. It was later revealed that GPT-5 did not actually solve these problems but instead identified existing solutions in the literature. After facing criticism from rivals like Yann LeCun and Google DeepMind CEO Demis Hassabis, Weil removed his initial post. OpenAI has now published supporting remarks from mathematicians Noga Alon, Melanie Wood, and Thomas Bloom, the latter of whom described Weil's earlier statement as "a dramatic misrepresentation." According to OpenAI, mathematicians have believed for nearly 80 years that the best solutions to these problems resembled square grids. The company stated, "An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better." This proof marks the first time AI has autonomously solved a significant open problem in mathematics. The proof was generated by a new general-purpose reasoning model, rather than a system specifically designed for mathematical tasks. OpenAI asserts that this development indicates AI systems are becoming more adept at connecting ideas across different fields, which could have implications for various disciplines including biology, physics, engineering, and medicine. Thomas Bloom remarked on AI's potential to explore the depths of mathematics, questioning, "What other unseen wonders are waiting in the wings?"

OpenAI makes breakthrough in 80-year-old math problem with 'ingenious ideas'

OpenAI claims its model solved a famous geometry problem that has eluded the world's greatest mathematicians for 80 years -- a breakthrough hailed as evidence of the bot's creativity and "intuition." The company published its findings on Wednesday, demonstrating that one of its models cracked the planar unit distance problem, first posed by legendary Hungarian mathematician Paul Erdős in 1946. The puzzle poses a deceptively simple question that boils down to: How many pairs of dots on a piece of paper can be the same distance apart? The prevailing theory suggested that a "square grid" was the key to creating the maximum number of pairs listed in the problem, and Erdős himself proposed that the number of pairs could increase only slightly faster than the number of dots as more points were added. OpenAI's work, however, disproved the idea and proposed its own layout. University of Toronto mathematician Arul Shankar, one of the experts who reviewed OpenAI's work, went as far as to suggest the model used its own "intuition" to arrive at the surprising solution. "In my opinion, this paper demonstrates that current AI models go beyond just helpers to human mathematicians - they are capable of having original, ingenious ideas, and then carrying them out to fruition," he said in a statement. Fellow Toronto professor Jacob Tsimerman said he was wowed by the results, noting that he had once tried to disprove the distance problem himself to no avail. "It is definitely an intimidating construction to see through, even if you know what is going on, and even harder to go play for yourself," he noted. OpenAI has repeatedly touted its model's use to help solve math problems and prove or disprove decades-old conjectures that were previously deemed too complicated to approach. The breakthrough came just days after it was revealed that OpenAI's ChatGPT was used to help solve another decades-old problem, allowing the mathematician who first thought up the solution to see his "sensational" idea proven right. In 1995, renowned French mathematician Michel Talagrand, 74, made a sweeping statement, claiming that in a seemingly endless and scattered plane littered with points across innumerable dimensions, simple, orderly shapes will appear. But what Talagrand believed would be a herculean task to prove or disprove his so-called "convexity conjecture" came to an abrupt end last week after mathematicians at the California Institute of Technology used OpenAI's ChatGPT to play out his theory. "This is the most extraordinary result of my entire life," Talagrand told Scientific America about seeing the answer. "The proper word is 'sensational.'" At the heart of Talagrand's conjecture was the idea that even when facing a billion dimensions, one could draw a simple shape that manages to encircle an enumerable number of points scattered throughout them. The French mathematician himself was the first to pour cold water on his own theory, describing it as a "shot in the dark" and saying that, if true, it would amount to nothing less than a "total miracle." Talagrand had even put up a $2,000 reward for years for anyone who could take on the challenge, but a collector never came. That is, until Antoine Song and his student, Dongming (Merrick) Hu, used ChatGPT to translate Talagrand's problem and demonstrate that he was right. The duo eventually worked with Stefan Tudose, a Princeton mathematician, on the final proof, choosing to exclude ChatGPT due to uncertainties in the language models' "thought process."

ChatGPT solves 80 year old math problem that puzzled researchers for years

This announcement is interesting because OpenAI faced criticism last year after making a similar claim. OpenAI has once again made a big claim in the world of AI and mathematics. The company says that one of its latest AI reasoning models has successfully solved a famous geometry problem that remained unsolved for nearly 80 years. The problem was originally proposed by legendary mathematician Paul Erdos in 1946. This announcement is especially interesting because OpenAI faced criticism last year after making a similar claim. About seven months ago, former OpenAI VP Kevin Weil posted on X that GPT-5 had solved several unsolved Erdos problems. However, it was later found that the AI had only rediscovered solutions that already existed in mathematical research papers. That earlier claim led to criticism from well-known AI researchers like Yann LeCun and Demis Hassabis. Weil later deleted his post. This time, OpenAI appears to have taken extra care before making the announcement public. The company also shared supporting comments from respected mathematicians, including Noga Alon, Melanie Wood and Thomas Bloom. Also read: OpenAI introduces Guaranteed Capacity offering: What is it and what it promises 'For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids,' OpenAI posted on X. 'An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better.' OpenAI says this is 'the first time AI has autonomously solved a prominent open problem central to a field of mathematics.' Also read: Google unveils new AI Ultra subscription with 20TB of cloud storage: What it offers and how much it costs The company also explained that the proof was generated by a general-purpose reasoning model rather than a system specially built for solving maths problems. According to OpenAI, this shows how modern AI systems are becoming better at handling long and difficult chains of reasoning. 'AI is helping us to more fully explore the cathedral of mathematics we have built over the centuries,' Bloom was quoted as saying in OpenAI's blogspot. 'What other unseen wonders are waiting in the wings?'