OpenAI model disproves famous math problem from 1946 that stumped mathematicians for 80 years

An OpenAI model solved a famous math problem that stumped humans for 80 years

In mid-May, OpenAI announced that an internal AI model had disproved the Erdős unit distance conjecture, a famous problem in discrete geometry that had stumped human mathematicians for the last 80 years. OpenAI gave several mathematicians early access to the result and published their reactions. Tim Gowers -- who won the Fields Medal, the most prestigious prize in mathematics -- wrote that "there is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics." University of Toronto professor Daniel Litt wrote that "this is the first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator." It's arguably the first time that an AI system has found a proof resolving a major open conjecture. That's impressive, but I don't view it as a radical break from the previous trajectory of AI progress in mathematics. Three years ago, LLMs struggled to solve arithmetic problems. It was only last year that LLMs started acing high school mathematics competitions. When I attended the Joint Mathematics Meetings -- the largest annual mathematics conference in the world -- in January, I learned that AI systems were starting to contribute to mathematical research, but only in constrained settings. It took significant human interpretation to turn an AI output into a publishable theorem. OpenAI's new result is the next step in this progression. The AI model cleverly applied existing ideas drawn from several subfields of mathematics to create a full proof. But it didn't pioneer any genuinely new techniques. The result has since been cleaned up and extended by human mathematicians. This points to a medium-term future where human mathematicians and AI models complement each other: AIs have a broader knowledge of past work than any human alive and much more willingness to grind through tedious proof strategies that aren't likely to work. But humans can still think more deeply about any one problem and ask more interesting questions. That might not last. AI systems have been improving at math so rapidly that it's unclear what role, if any, human mathematicians will play a decade from now. The unit distance problem Paul Erdős was one of the most prolific mathematicians in history. He wrote over 1,500 papers in his lifetime, the most ever. One of his greatest talents was coming up with problems that are simple to state but have deep roots. In 1946, he introduced the unit distance problem. Imagine you have some points in a 2D plane and you measure the distance between each pair of points: In this diagram, there are five points and ten pairs of points. Three pairs happen to be exactly 1 unit apart: AD, BE, and CE. Can we rearrange the points so that more pairs of points are exactly 1 unit apart? Yes. For instance, we could move points A and D to be closer to the B, C, and E cluster. With a bit more work, we could further rearrange the points so that there are seven pairs exactly one unit apart. But that's the most we can do. We could do the same analysis with 6 points, 7 points, and so on. But as the number of points grows, the problem very quickly becomes too complicated to find the exact answer. So instead of asking exactly how many unit distances are possible for a given number of points, Erdős tried to calculate upper and lower bounds on the number of length-one lines for n points, assuming that n is a large number. To help calculate a lower bound, Erdős assumed that the points would be laid out in a grid. This is probably not the optimal layout, but if he could demonstrate that points in a grid have a certain number of pairs with unit distance, then the optimal arrangement must have at least that number. The simplest option is to space the grid so that every point is distance 1 from its neighbors directly above, below, left, and right. However, Erdős saw that you could do even better if you took diagonals into account. If you make the grid spacing smaller, you can make each point be distance 1 from a greater number of neighbors. In the diagram above, if the grid spacing is 1, then each individual point is one unit away from four neighbors (the left panel). Instead, if the grid spacing is ⅕ (as shown on the right), then each individual point is one unit away from 12 neighbors: OpenAI's write-up of its new result included a confusing diagram showing points in a grid with a bunch of lines connecting them. The diagram becomes easier to understand if we superimpose a circle like this: This works because of the Pythagorean theorem, which states that if we have a point that is a units to the right and b units above another point, the distance c between those two points satisfies a² + b² = c². The trick is to choose some number c² so that there are a whole bunch of pairs of whole numbers a and b such that a² + b² = c². Then, if we scale the grid down so that each point is 1/c from its neighbors, there will be a bunch of unit distances. For example, if we choose c² = 25, then the Pythagorean equation can be satisfied by either 0² + 5² = 25 or 3² + 4² = 25. This corresponds to the 12-grid-point circle I showed earlier, with points at (0,5), (3,4), (4,3), (5,0), (-4,3), (-3,4), and so forth. (Technically, these lengths should all be divided by 5 -- (⅗, ⅘) for example -- but I'm leaving the denominators out for clarity.) OpenAI's diagram is based on choosing c² = 65, which can be satisfied by either 1² + 8² = 65 or 4² + 7² = 65. This means that if the grid spacing is 1/√65, each point will be one unit away from 16 other points: (1,8), (4,7), (7,4), (8,1), (-1,8), (-4,7), and so forth. Larger values for c² -- if they're chosen carefully -- enable more whole-number diagonals and hence more unit-distance pairs. However, if c² is too large compared to the number of points in the grid, then many of the potential one-unit-away neighbors will be outside the grid. In short, we want to choose a c² that's large enough but not too large. Using insights from number theory, including Jacobi's two-square theorem, Erdős was able to show that an optimally sized circle will enable the number of unit-distance pairs to grow faster than the number of points, but only barely. The question became "can you do better?" To find an upper bound, Erdős used an argument from a quite different area of mathematics called graph theory to show that you could only have so many unit distances. But his upper bound grows much, much faster than the best lower bound he was able to construct. Erdős's conjecture was that the actual optimum was much closer to the lower bound than the upper one. He predicted, but couldn't prove, that the maximum number of unit-distance pairs grows just barely faster than the number of points. To be more precise, Erdős conjectured that the number of unit distances would be n^(1+o(1)). In other words, for a sufficiently large n, the maximum number of unit distances would be less than n^(1+𝜖) for any 𝜖 > 0. That could end up growing a little faster than his lower-bound construction -- which was n^(1 + C/(log log n)) for some constant C -- but within the same general ballpark. Proving his guess became known as the unit distance problem. For the next 80 years, it looked like Erdős was right. Then an OpenAI model proved him wrong. The AI's approach Erdős's conjecture assumed that, at least for a large number of points, a square grid could yield about as many unit-distance pairs as organizing the points in other ways. OpenAI's AI proved this wrong by demonstrating that there was another, more complex way to organize n points that allowed more pairs to be exactly one unit apart. Precisely because the new pattern of points is more complicated, it's tricky to explain it concisely. But you can think of it as a clever modification of Erdős's grid. The AI constructed a grid in a high-dimensional space and then projected this more complex structure into two dimensions. And instead of using a whole-number grid with points like (1,3) or (-3,6), the AI construction used something called algebraic integers to build this more complicated grid. It turns out that this kind of higher-dimensional grid has richer structure, which allows the AI to pack more unit distances into the same number of points. It's hard to illustrate this alternative arrangement of points because it only becomes advantageous with a very large number of points. But here's a simpler arrangement of points that was constructed in a similar way. You can click here if you want to play with the illustration yourself. It has 1,345 points and only produces 5,916 unit distances, fewer than the 7,632 unit distances that a square 1,296-point grid produces using the Erdős technique. But I think it gives a sense of how a pattern that isn't a grid could produce more unit distances than a square grid. The more complicated patterns pay off. While the OpenAI model's proof does not explicitly state how many unit-distance pairs are possible for n points, human mathematician Will Sawin was able to show that it grows at least at the rate of n1.014. This might seem small, but as n gets really big, this number will become much larger than the counts produced by the Erdős approach. That being said, the AI's result doesn't completely resolve the problem. Our best upper bound for the number of unit distances is around n1.333. More work is needed to close this gap. How does this result fit into AI for mathematics? If you'd asked me two weeks ago -- before OpenAI's announcement -- about the most novel contributions of LLMs to mathematics, I probably would have pointed to the AlphaEvolve system from Google DeepMind. AlphaEvolve harnesses LLMs to be the engine of an optimization process. If you can turn a math problem into a piece of code to optimize, which you often can, the LLM might find better solutions than humans have for certain types of problems. In November, four mathematicians (including Terence Tao) released a paper that analyzed AlphaEvolve's performance on 67 optimization problems across the mathematical literature. They found that AlphaEvolve was able to improve on the established literature in some cases. This was a step up in autonomy from previous LLM contributions, such as literature review, but it still required humans to frame it as an optimization problem and turn the AI's output into usable mathematics. And only certain types of problems are amenable to this approach. More conceptual questions that don't include a number to optimize can't easily be studied with AlphaEvolve. So AI companies have been working to develop LLM systems that can directly output a correct solution to any math problem. OpenAI's result is a substantial step in that direction. But it also fits the pattern of previous AI-assisted mathematics. For one thing, other companies have also worked to solve Erdős problems. Because Erdős posed hundreds of problems over his career -- and because mathematician Thomas Bloom has organized an effort to compile all of them at www.erdosproblems.com -- AI companies have used them as a testing ground to evaluate AI systems. In January, Cambridge undergraduate Kevin Barreto worked with a friend to ask GPT-5.2 and Harmonic's Aristotle to produce the first autonomous solution of an Erdős problem. On May 22, two days after OpenAI's announcement, Google announced that its AI system had solved nine open Erdős problems, including two that had been open for over 50 years. To be clear, the problem that OpenAI solved is more impressive than any of the other work I just mentioned. But OpenAI's solution is more in line with past AI efforts than the headline result might suggest. One reason the unit distance problem was unsolved for 80 years, despite being so well known, is that most people thought Erdős's conjecture was true. But the mathematical tools we have are nowhere close to being able to prove Erdős's bound. So mathematicians expected that any proof of the conjecture would involve major new ideas or approaches. Instead, as we've seen, the AI disproved the conjecture by making an extension of Erdős's initial construction. It was a clever and nonobvious solution, but it also bore some similarity to the kind of optimization work done by a system like AlphaEvolve. This dynamic is reflected in some of the mathematicians' responses. Mathematician Tim Gowers wrote that when he first heard about the AI's result, he thought it had proved the theorem. "I spent the evening adjusting my world view: If the AI could come up with a proof like that, then maybe it would be all over for mathematicians very soon." But the next morning, Gowers and other external reviewers received an email about the result, and he realized that the LLM "had disproved the conjecture rather than proving it, which came as a big relief." OpenAI's solution also had two properties that played to the strengths of AI models relative to humans. First, the eventual solution relied on applying sophisticated techniques from a quite different area of mathematics: algebraic number theory. AI systems have been trained on huge swaths of mathematics -- and there's a lot of math out there -- so they have a broader knowledge of previous mathematical work than any human in the world. For a human to solve this, they would have needed to have the relevant algebraic number theory knowledge while also being interested in the unit distance problem, a rare combination. Second, the reasoning process was such a grind, and seemingly unlikely to succeed, that most humans would not have thought it worth the trouble. Jacob Tsimerman, a University of Toronto professor, remarked in the OpenAI document that he had briefly considered taking a similar approach to disprove the conjecture. But that type of technique "consumes much time and frequently doesn't work out," so he abandoned the project. An AI, on the other hand, can work through many proof strategies that don't work out before discovering one that does. OpenAI could have run the problem many times before a model found a solution. Indeed, an OpenAI chart revealed that even with the maximum token budget, the internal model solves the problem only half of the time. To be clear, what the AI system did is still impressive. "It's always tempting to look at a completed proof and declare it obvious after the fact," Tsimerman said later in his remark. But as I noted previously, it also played to the strengths of AI systems. In the short to medium term, this points to a world where AI models complement humans but do not replace them. AI systems will tackle lists of problems curated by human mathematicians or aid humans in finding relevant approaches from seemingly unrelated mathematical fields. But they won't immediately displace the human role in choosing which questions to ask or developing wholly new techniques. Even this result was very much a human-AI collaboration. While the AI system found the proof on its own, human mathematicians verified the result. Other humans came up with better-written proofs that extended the AI's initial ideas, like Will Sawin finding an explicit lower bound as I mentioned above. It's unclear how long this complementarity will last, however. Gowers spent the rest of his comment exploring whether the relief he felt on hearing that AI had disproved the conjecture was justified. He more or less concluded that it was, but in a footnote, he wrote that he would guess "that AI will soon reach a high level at other activities such as building theories, formulating definitions and asking interesting questions." In the past year, we've gone from AI systems that hadn't yet beaten high school mathematics competitions to ones that can advance mathematics in interesting ways. It seems likely that AI systems will continue to become more autonomous when working on mathematical problems. At the same time, we haven't fully explored what current models can achieve in math. Soon after OpenAI's announcement, University of Michigan postdoc Xiao Ma found that GPT-5.5 was also able to prove Erdős wrong if given a small hint. If a generally available model could disprove this famous conjecture and no one noticed, what other discoveries could happen today that no one has thought to try? Kai Williams is a reporter for Understanding AI, a Substack newsletter founded by Ars Technica alum Timothy B. Lee. His work is supported by a Tarbell Fellowship. Subscribe to Understanding AI to get more from Tim and Kai.

An OpenAI Model 'Disproved' a Famous Math Conjecture. This Mathematician Couldn't Leave It Alone

Will Sawin got OpenAI's email on a Friday night. Or Saturday morning. Either way, Sawin, a professional mathematician, spent his entire weekend thinking about that email. By next Monday, he decided to write up a paper that essentially improved what was given to him -- an AI's "proof" of Paul Erdős's unit-distance problem, an infamous conjecture from 1946. Last week, OpenAI published a blog post on the AI's proof. The paper came with a companion piece containing comments from nine renowned mathematicians uninvolved with OpenAI, including Sawin. Many prominent mathematicians praised the work, with Fields medalist Tim Gowers calling it a "milestone in AI mathematics." This result is just one of dozens of AI-derived solutions to long-time mathematical riddles. All this has us asking: Could AI usher in a new era of mathematical advancements? The answer, if one even exists, is likely a nuanced one. There are certainly computational advantages that AI brings to the equation (no pun intended). But what does this really mean? Does that represent some tangible revolution, or is it a "misconception" stemming from AI's data-driven imitation of human intelligence, to quote Pope Leo's recent encyclical? We'll for sure continue to see more AI solutions pop up -- after all, impossible math conjectures come by the hundreds -- and each time, human mathematicians will be summoned to check the computer's work. To OpenAI's credit, its blog post closes with this pleasant sentiment: "People choose the problems that matter, interpret the results, and decide what questions to pursue next." So, Gizmodo reached out to Will Sawin, who appears to have done just that: interpret the results and pursue a relevant question. We wanted to know what the experience was like. Sawin is a Fernholz Professor of mathematics at Princeton University. He began his academic career at Yale when he was 10 years old and has since worked in number theory, algebraic geometry, and combinatorics. During the conversation, Gizmodo asked Sawin about his own experience reviewing AI-derived mathematical proofs, the reality of using AI in mathematics -- and, most importantly, what it is and is not doing. The following conversation has been edited for grammar and clarity. Gayoung Lee, Gizmodo: Let's start with the headline of OpenAI's blog post: "an OpenAI model has disproved a central conjecture in discrete geometry." What's the conjecture we're talking about here? Will Sawin: We start with the problem in simple form: if you have a set of points in the plane, how many pairs of points can have a distance exactly 1 [unit distance] from each other? You can play around with different constructions. If you try it by hand, the best you're going to find is some kind of grid, like a triangular grid of points. Each point will be the next unit distance from 6 other points. That's pretty good for small numbers of points. The mathematical question is for each n number of points, "What's the greatest number of pairs of unit distances you can get with that many points?" For Erdős, the problem was, how does this grow as a function of n? And the conjecture he made was that it can grow slower than every power of n > 1. So, slower than n, and slower than n, and so on. This is a purely asymptotic question, so not about any particular value of n. I think one thing that people were disappointed by is that OpenAI's paper, our paper explaining the proof, and my paper on an optimized version -- none of these papers had an example of the construction for a particular value of n. That's because of the asymptotic nature of the problem. When Erdős first brought up the problem 80 years ago, he was not sure it should be true. But he seems to have gotten a little more confident in this over time, as nobody figured out a way to make it grow faster than that. There are some conjectures in mathematics that, if you disproved them, it would be a really big shock to the mathematical community. It wasn't such a huge shock that this statement wasn't true, but it was the opposite of what people generally believed. Gizmodo: You mentioned that Erdős grew more confident in his question. To ask a more metaphysical question -- what does it mean to prove or disprove a conjecture in mathematics? What does it mean exactly for a problem to be "solved"? Sawin: To answer your first question, a proof or a disproof in mathematics is an argument that is completely convincing, leaving no room for doubt. What mathematicians have considered to be a proof has changed over time. Hundreds of years ago, you might see people use some kind of physical reasoning and consider that to be about the real world and consider that to be a mathematical proof. Now people have different standards. One standard is a formal proof. There exist formal proof systems with logical rules for what statement you're allowed to introduce from another statement. One common belief is that a proof really should be a formal proof. If you have an informal proof using English words, it's only a valid proof if it's an explanation of why there exists a formal proof. Other people disagree and say there's something about informal proofs that is not completely captured by formal proofs. Let me not take a position on that philosophical question right now (laughs). But the [OpenAI] proof in question is an informal proof that you could eventually turn into a formal proof if you wanted. And this is usually what mathematicians mean when they say something's been "proved." Gizmodo: So OpenAI's proof is an informal proof? Sawin: It's an informal proof that looks very similar to the informal proofs that mathematicians produce. There are some things about the way it's organized that, if you've seen a lot of mathematical proofs, you can tell aren't exactly the same as how a mathematician would write them. But I would say someone who has not read a lot of mathematical proofs might not be able to tell the difference. Gizmodo: Could you unpack for me the content of this informal proof? How did the AI arrive at its conclusions, from what you can tell? Sawin: If you were trying to prove what Erdős said -- an upper bound for the number of unit distances -- you'd have to reason about any possible collection of points on the plane to make some argument that's valid for any set of points on the plane. Which would be hard. So the disproof, in some sense, is easier, because you have to come up with a specific sequence of points in the plane. So it's a question about the limit as the size goes to infinity, and you need to construct a sequence and show it has a lot of unit distances. The way the AI did this was to use algebraic number theory to use a ring of integers in an algebraic number field. When I describe this idea at that level of generality, it's not so foreign to what mathematicians had already tried. I would say the key thing that AI realized that humans didn't is not just that you can use algebraic number fields but that you can use algebraic number fields of growing degrees. You let the degree of the field grow, which basically increases the kind of complexity of the numbers you're working with. And that makes the number of unit distances grow very rapidly and more rapidly than Erdős expected. Gizmodo: I'm admittedly somewhat skeptical about news that AI did this impossible thing in math, since my thought tends to go to how AI is just a really great computing device. But here, it sounds like OpenAI's model looked at what human mathematicians were doing and kind of... made a logical decision as opposed to pure computation. Is that something we've seen before? Sawin: It depends on what you mean by "before." In terms of the cleverness of the mathematical reasoning, there's not a big gap between this and the most impressive previous cases of OpenAI and mathematics that I have seen. Erdős had a huge number of problems, and Thomas Bloom collected over a thousand on a website. And he has not collected all the problems that Erdős asked; there are even more than that. Over the last few months, a lot of people have been trying to use AI to generate solutions to Erdős problems. Some solutions aren't incredibly impressive. Some of them are, like, the AI discovers there's a paper where somebody solved the problem already, but they just didn't know about the problem. So they didn't know, but the [AI realizes] their paper just immediately solves the problem. But in some cases, AI introduces some new ideas that humans didn't use. Sometimes, the idea isn't very interesting. Other times, the idea is interesting and leads humans to wonder what they could do with this idea. That is not too dissimilar to what happened here [with OpenAI]. I think this was a more technically intricate idea than previous problems that I've seen. It definitely was a bigger problem that more people knew about and more people had worked on. I can tell you reasons for skepticism. So certainly, this is an idea that, as far as we can tell, humans did not come up with. This is not an idea that humans couldn't have come up with. I can see why it was hard for somebody to come up with that idea, but people come up with ideas that are hard for people to come up with all the time. It's definitely not a situation where AI is doing something that humans can't do. We don't know the amount of computing power that was used, the amount of problems that OpenAI tried it on, or how the AI system is set up. I mean, OpenAI did say some things about this, but certainly not at the level of detail that they used to have when AI was not such a big deal. So exactly how hard it is for the AI to do this is something we don't fully know. Gizmodo: On that note, let's talk about your own preprint. This "refined" the AI's proof. What does that mean? Sawin: Yeah. So the conjecture is that this function grows slower than any power of n > 1. The OpenAI proof said that it actually grows faster than a power of n > 1. But it doesn't tell you which power of n > 1, just "some" power of n > 1. I wanted to try getting an explicit value that's reasonably good, that's not a really tiny distance away from 1. The value I ended up getting is a little more than 1.01, so not a big difference from 1. Other people have improved it, and it's now more like 1.03. And as I was reading the argument of the AI and trying to understand each step, I was thinking, 'How would I do each step in an efficient way?' A lot of things about the AI's argument were inefficient in some way, which I think is not surprising. Like, a human wouldn't have written it that way. But the AI was clearly not trying to be efficient. There was definitely nothing wrong with the original argument. It achieved this goal. I had a somewhat different goal. But basically, almost every step of the argument had to be changed in some way to support this goal of getting a reasonable, explicit constant. Gizmodo: And OpenAI was fine that you did this? Sawin: Oh yeah, I told them. And they were fine with it. Their announcement mentioned my paper. No concerns. Gizmodo: Has this experience influenced at all how you view AI's impact on your work, or the perception of your work, as a mathematician? Sawin: There are two things that these AIs are very effective at now. One is searching the literature. If you want to know if a certain theorem exists, you're much more likely to get the result by asking an LLM than you are by typing it into a mathematics-specific search engine. The other thing it's good at is reading and proofreading a paper -- not just typos or grammatical errors but also mathematical errors that could affect the result. I mean, I do still read and proofread my paper as a human, because, you know, that's what's appropriate. And I think most other mathematicians don't want to let it write their papers for them, either, for reasons of personal credibility and accountability, but also because one still shouldn't fully trust the AI. They've gotten a lot better, but there are still blind spots. If you're using an AI-generated idea, you should still express your understanding of it as a human. As for generating ideas, I haven't found AI very effective at that. I think that's partly because it varies from one field of mathematics to another. It's easier for AI to generate new ideas in combinatorics than in some areas where there's more technical background and fewer prior works to use as a point of comparison. Gizmodo: And as you mentioned, sometimes it's because an AI having an idea doesn't necessarily mean that it's an idea that humans would never have thought about. Sawin: Yeah. Like, if I were to ask AI for an idea, I wouldn't be asking so I could throw out the ones I already came up with for a question. Gizmodo: So to you, a mathematician, is AI a partner or a search engine tool? Sawin: Currently, I'd say tool. The people that are having the most success are using AI as a tool. It's completely conceivable that soon somebody will prompt an AI to look at the recent literature, come up with its own problems, and solve those problems. And it'll come up with interesting stuff. But that hasn't happened yet, as far as I'm aware. I don't want to try to make any predictions about the future, but that's what's true currently. Gizmodo: Okay. So I'd like to ask you for some advice. When we see another AI-written, fantastical proof, what do you think is the most important question we should ask ourselves? Sawin: I'd say that it is probably less exciting than it sounds (laughs). But it's probably still somewhat exciting. If somebody's like, "Oh, AI solved this impossible math problem," it's definitely not that. It's not that AI solved an impossible math problem, but it's not nothing. It's somewhere in between. I've definitely seen people whose first reaction is to assume that if AI solved an impossible math problem, like, human mathematics is over. And I've definitely seen people whose reaction is to assume that it's nothing, that AI really didn't do anything. It's definitely somewhere in between those two. And I'm not sure exactly where it is, but it's somewhere in between those two.