设$latex C>0$，$latex\epsilon>0$足够小。是否有无限多个整数$latex a、b、n$，其中$latex为a \geq \epsilon n$，$latex b \geq \epsilon n$，使得$latex a！b！\mid n！（a + b – n）！$ $latex a + b > n + C \log n$？
5月20日，OpenAI 分享了单位距离问题的解决方案：https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf，以及一篇博客文章：https://openai.com/index/model-disproves-discrete-geometry-conjecture/ 解释了这项工作，还有一篇配套论文：https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf，由九位世界级数学家评论了证明的正确性及工作的意义（同时呈现了一个简化的人类版本结果）。数学家们普遍认为厄尔多什的猜想——关于在平面上可以放置多少均匀分布的点——是正确的。出乎意料的是，OpenAI 的内部模型找到了一个反例。为此，它找到了一种复杂的方法来使用一种称为代数数论的数学领域的工具。正如多伦多大学的 Jacob Tsimerman：https://www.math.utoronto.ca/~jacobt 在配套文章中写道，“这是一项非常令人印象深刻的工作……这绝对是一个令人敬畏的构造。”
在同一篇文章中，剑桥大学和法国科学院的 Tim Gowers（https://www.college-de-france.fr/en/chair/timothy-gowers-combinatorics-statutory-chair/biography）写道：“如果这篇论文是由人类撰写并提交给《数学年刊》，而我被要求给一个快速意见，我会毫不犹豫地推荐接受。此前没有任何 AI 生成的证明能接近这一水平。”
在提出原始解法的论文中，作者仅被记为“OpenAI”。
特伦斯·陶 (Terence Tao) 遇到埃尔德什 (Erdős) 时年仅 10 岁。
后来，一组包括 Bloom 在内的四位数学家，使用与 AI 模型应用于单位距离问题相关的技术，反驳了（https://arxiv.org/abs/2605.28781）另一长期存在的埃尔德什猜想的一个版本。“和-积”猜想（https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/）提出，如果你有一组数字，它们的和或积必须快速增长。数学家们找到了一个实数集合，使得其和与积的增长都比预期慢。对于整数的猜想仍然悬而未决。
了解 AI 对数学及数学家的影响，不仅要看它的最重要成果，还要考察它如何改变日常解决普通问题的实践。普林斯顿大学数学家诺加·阿隆（Noga Alon）估计，在他的职业生涯中，他已经解决了几十个埃尔德什问题。他现在已经停止尝试。“一旦 AI 开始解决这些问题，再也没有意义了，”他说。特里·陶 (Terry Tao) 已远离埃尔德什问题社区，专注于完成自己的工作。
An editorially independent publication supported by the Simons Foundation.
Create a reading list by clicking the Read Later icon next to the articles you wish to save.
Type search term(s) and press enter
O n May 20, 2026, OpenAI made an announcement that shook the mathematical world. An internal AI model — one not available to the public — had come up with a counterexample to the “unit distance” problem：https://openai.com/index/model-disproves-discrete-geometry-conjecture/, a conjecture made in 1946 by Paul Erdős, the prolific, itinerant Hungarian mathematician.
Erdős posed thousands of questions, but this one was special: It was both simple to explain and mathematically deep. It was the first historically significant proof to come from an AI model. Though the model’s result wasn’t definitive — human mathematicians would substantially improve on it within weeks — it was innovative, bringing in ideas from a distant branch of math that no one had successfully applied to this problem before. And it was influential: Within a few days, related techniques were used to solve other important problems.
Then on August 1, OpenAI announced：https://openai.com/index/ten-advances-in-mathematics/ that an unreleased model named Astra made 10 additional mathematical advances, including finding solutions to three more problems posed by Erdős.
Many mathematicians have hailed developments such as these as a phase transition in the mathematical capability of AI models. These models are “changing dramatically the way mathematical research is being done,” said Noga Alon：https://web.math.princeton.edu/~nalon/ of Princeton University, who has solved dozens of Erdős problems over his decades-long career.
Paul Erdős, one of the most prolific mathematicians in history, was deeply whimsical when it came to mathematics, and deeply cynical when it came to authority.
Erdős and his conjectures have long fascinated mathematicians. He traveled constantly — living out of a suitcase for years at a time, staying with friends, owning almost nothing. He rattled off problems in published papers and letters to mathematicians around the world, often attaching prize money that he would pay out of pocket to the first person to come up with a solution. The reward might be a token $10 or $25, or, for problems he considered important or difficult, it could range into the thousands. Erdős died of a heart attack in 1996 while attending a math conference in Warsaw, but a nonprofit foundation based in Iowa has promised：https://www.combinatoricsfoundation.org/erd%C5%91s-problems to make good on his bounties.
He was a beloved figure, but also a downright weird one. He only wore silk, and he avoided the touch of other people. Deeply cynical about authority, he gave away most of the money he earned and relied on a friend to manage his finances and other practical affairs. He referred to God as the “Supreme Fascist” and fueled his incessant output of mathematical ideas with a steady diet of amphetamines. It is a strange irony of history that the problems he suggested have now become a central proving ground — and, in effect, a series of PR coups — for the world’s biggest and most powerful technology companies.
But in all likelihood none of this would have happened had it not been for an English mathematician named Thomas Bloom：http://thomasbloom.org/.
Like Erdős, Bloom was interested in both number theory and combinatorics. His focus has been an area called arithmetic combinatorics, which lies at the intersection of the two. After getting his doctorate in 2014, Bloom established himself as a rising star in the field, landing a prestigious fellowship from Britain’s Royal Society：https://royalsociety.org/grants/university-research/, which let him work at almost any university he wanted to. (He’s now at the University of Manchester.)
Bloom has liked Erdős’ style for as long as he can remember. But he always found it hard to keep track of which problems had been solved and which had been forgotten entirely. So in early 2023, he decided to gather as many problems as he could into a list.
He intended it for his own use. But “I thought it would be easier if I could access it wherever I was,” he said; he figured he “might as well make a website, kind of with the expectation that maybe nobody would use it.” He gathered a couple hundred problems and launched erdosproblems.com：http://erdosproblems.com. Bloom used ChatGPT to write the Python code that ran the website, which was, at the time, a remarkable thing for a large language model to be able to do. Using one to collaborate on the math itself still seemed like only a distant possibility.
Thomas Bloom’s website of Erdős problems became a home for mathematics at its best. Then AI came on the scene.
His goal was not just to cross items off a list. He wondered if “modern day mathematics, often using techniques unknown by Erdős, could clear up many of these more obscure problems,” he wrote in a blog post：https://www.erdosproblems.com/forum/thread/blog:1. “We will then be left with a core of interesting, difficult problems, which can serve to demonstrate the limits of our knowledge.”
Bloom did crucial work in curating the list: Sometimes Erdős stated problems in ambiguous or unclear ways, and Bloom figured out what the most sensible version of each problem should be. He kept adding problems to the site, and gradually its audience grew. Over the course of 2024 and the first eight months of 2025, the statuses of 111 problems on the list were changed from “open” to “solved” (although some of these had been solved years earlier, and their status change reflected the rediscovery or verification of a proof).
Then, in August 2025, some colleagues suggested that Bloom add a commenting function, so that people could talk about problems they were interested in. He was able to do so quickly, using ChatGPT to write the code. By now he’d cataloged nearly 1,000 problems.
Bloom’s timing was good. He made it possible for like-minded people to talk to one another, and that “really let a community build up,” he said. For the most part, comments were sporadic — a problem might attract a single comment pointing out an example or noting how hard the problem looked. But activity steadily grew, and some problems catalyzed nuanced mathematical discussions between strangers.
“Tom probably never really realized this, but for me it’s honestly changed my life,” said Wouter van Doorn, the fourth-most-prolific commenter：https://www.erdosproblems.com/forum/user/Woett?all_posts=1 on Bloom’s website. Like many people who became active on the site in the autumn of 2025, van Doorn isn’t exactly a professional mathematician. He works “for a company that gets hired by other companies to do customer service support,” as he put it. But he isn’t exactly an amateur either — a decade prior, he almost completed a master’s degree in math at KU Leuven in Belgium. In 2024, spurred in part by how capable he saw LLMs getting, he took a six-month leave of absence from work to focus on math. At the time, while he didn’t particularly want to use AI, he remembers thinking, “Right now I’m still better at mathematics than an AI is, but who knows what it’ll be in a year, two years, five years? If I want to finish these projects, and I want them to be mine, now is the time.”
We say that $latex A \subseteq \mathbb{N}$ has property $latex P$ if, for all $latex n \geq 1$, there are only finitely many $latex a \in A$ such that $latex n + a$ is square-free. We say that $latex A$ has property $latex Q$ if there are infinitely many $latex n$ such that $latex n + a$ is square-free for all $latex a < n$. How fast must sequences $latex A = \{a_1 < a_2 < \cdots\}$ with properties $latex P$ or $latex Q$ increase?
And so, in October 2025, van Doorn, now back at his day job, left the first comment on the page for Problem 1102：https://www.erdosproblems.com/forum/thread/1102?order=oldest. The problem, which Erdős posed in 1981, asks about properties of sets of “square-free” integers — that is, integers that have no repeated prime factors. (For instance, 30 is square-free because it is equal to 2 × 3 × 5, but 18 is not, because it is equal to 2 × 3 × 3; the 3 repeats.)
In early November, van Doorn shared progress toward an answer — which he’d figured out without relying on AI — as a comment on the problem page.
Later that day, another commenter on the site replied, claiming he had found a flaw in van Doorn’s argument. The two traded remarks in rapid succession, and van Doorn convinced his interlocutor that his argument was correct. “I see how your argument works now. Nice!” the other mathematician replied. That other mathematician was Terence Tao：https://www.math.ucla.edu/~tao/, a professor at the University of California, Los Angeles who is arguably the best-known mathematician alive today, and inarguably one of the most influential. (Not incidentally, when Tao was just 10 years old：https://blogs.ams.org/blogonmathblogs/2015/09/29/that-time-terrence-tao-won-500-from-paul-erdos/, he crossed paths with Erdős.)
Bloom’s website, which has the look and feel of an earlier time, was becoming an example of the internet at its democratic best. “This entire collaboration would not have been possible without Tom’s website and the comments section there,” van Doorn said. It didn’t matter if you had tenure or not, if you were young or old, if you were at a fancy university or even at a university at all. If you wanted to work on math and had good ideas, you could find people to collaborate with.
But as the winter set in — around the same time that van Doorn found himself collaborating with Terry Tao — things started to change.
Kevin Barreto and Liam Price, both in their early 20s, became friends in the summer of 2025 on a Discord server dedicated to AI. Barreto is currently an undergraduate at the University of Cambridge; Price studied some math in college but left before finishing. In December, convinced that the newest AI models might succeed in resolving some Erdős problems, the pair started throwing batches of problems at them. They realized early on that if they told GPT-5.2 that a problem’s answer wasn’t known, it wouldn’t make much headway, so as Barreto put it, they learned how to “prompt it in a very particular way, gaslighting it into thinking the problem is easier than it actually is.”
Let $latex A \subseteq \mathbb{N}$ be a set of density zero. Does there exist a $latex B$ such that $latex A \subseteq B + B$ and $latex |B \cap \{1, \ldots, N\}| = o(N^{1/2})$ for all large $latex N$?
They had what they thought was their first triumph on Erdős Problem 333：https://www.erdosproblems.com/forum/thread/333?order=oldest. Early on Christmas morning, Barreto posted a proof to Bloom’s website, writing, “We believe, to the best of our knowledge, this is the first case of an LLM fully autonomously resolving an Erdős problem, not previously resolved by humans.” Even though 333, which dealt with the sums of sets of integers, was not a particularly important problem, solving it with AI still felt important.
But a few hours later, another user pointed out that Erdős himself had provided a resolution to 333 in a paper published in 1977：https://www.sciencedirect.com/science/article/pii/0022314X77900038. Barreto owned up to the mistake. “My formal request to all members of the website is to put greater focus on literature search on the problems currently marked as open,” he wrote. “As someone who has fallen for this twice now, it’s quite gut-wrenching.”
Let $latex C > 0$ and $latex \epsilon > 0$ be sufficiently small. Are there infinitely many integers $latex a, b, n$ with $latex a \geq \epsilon n$ and $latex b \geq \epsilon n$ such that $latex a!b! \mid n!(a + b – n)!$ and $latex a + b > n + C \log n$?
Undeterred, he and Price kept at it, and by January 4, 2026, they’d used GPT-5.2 Pro to find a solution to Erdős 728：https://www.erdosproblems.com/forum/thread/728?order=oldest, a problem about when certain numbers are divisible by other numbers. This time nobody could find prior work already proving it. Barreto used another AI tool called Aristotle (developed by a startup called Harmonic) to certify that the proof held together logically. Nat Sothanaphan, a software engineer and the only forum participant more prolific than Bloom, Tao, and van Doorn, had ChatGPT write up the formalized result：https://arxiv.org/abs/2601.07421 and posted it online.
Price developed a methodology for how to ask LLMs to solve open questions. First, he would ask a chatbot for a solution. Then he would feed that solution into a fresh instance of the chatbot, asking it to check the previous chatbot’s work. He’d repeat this process until he had what looked like a workable solution. (This echoes some of the work that companies have been doing internally to create what they call harnesses or scaffolds, which automate the sort of iteration that Price does by hand.)
Barreto and Price’s papers represent just a fraction of the many Erdős problems solved at least in part by AI over the past few months. There are multiple reasons why these problems in particular have become such a fertile test bed for LLMs. The primary one is that, by and large, Erdős problems are in number theory, combinatorics, and graph theory, all areas of math that have proved more accessible than others to large language models. The problems also vary widely in difficulty and mathematical significance. This variation makes them appropriate for a nascent technology whose abilities also vary widely.
Many of Erdős’ problems had a monetary value attached to them from their moment of inception, a playful incentive from a wandering eccentric. But now, as the problems have become an informal benchmark for AI, their solutions are being discussed in terms of their “per-problem cost” — the price of the tokens needed to solve them.
“A lot of my recent papers should be mostly credited to AI,” van Doorn said. “The ideas involved were ideas I did not come up with myself.” Like many people active on the Erdős site, van Doorn is excited about the way LLMs are allowing him to do more things more quickly. “If I read an idea by an LLM, I digest it, try to understand it, simplify it, and generalize it,” he said. He uses AI to better understand the math.
Not everyone holds themselves to this standard. “A big problem is AI is being used a lot by people who aren’t mathematicians, who don’t have a huge mathematical background and are not capable of verifying the output,” Bloom said. “They like to move fast, ask their AI to check it, it grows and grows. We’re seeing a lot more of these 100- to 200-page papers that people are posting. ‘I solved this theorem; I got AI to generate the proof and check the proof and write the paper.’ But no human has read it, and no human is going to read it. It’s a huge challenge now.”
Is it true that, for any $latex x$, if $latex A \subset [x, \infty)$ is a primitive set of integers (so that no distinct elements of $latex A$ divide each other) then $latex \displaystyle\sum_{a \in A} \frac{1}{a \log a} < 1 + o(1),$ where the $latex o(1)$ term $latex \to 0$ as $latex x \to \infty$?
By Price’s own assessment, he doesn’t have enough mathematical understanding to verify the solutions he ultimately coaxes from the LLMs. But with Barreto’s help, he’s been able to find mathematicians knowledgeable and willing enough to check the results. Both Price and Barreto are co-authors with Tao, Jared Duker Lichtman：https://mathematics.stanford.edu/people/jared-duker-lichtman of Stanford University, and other accomplished mathematicians on a May 2026 paper resolving Erdős Problem 1196：https://arxiv.org/abs/2605.00301, one of their more significant results. (1196 asks about the possible size of so-called primitive sets — collections of integers, such as {2, 5, 9, 21}, in which no number divides any other.)
Bloom was surprised that despite lots of attention from OpenAI, Google DeepMind, and several startups, most of the new results had come from hobbyists and undergraduates using publicly available LLMs, not from corporate labs using more advanced internal models.
But that would change a few weeks later, on May 20, 2026, when OpenAI announced that they had solved one of the most well known Erdős problems of all, the unit distance problem.
In the first months of 2026, the major tech companies began to see opportunity in erdosproblems.com. As Lichtman explained, “Erdős had over 1,000 papers. They were scattered.” An institute in Hungary had collected scanned images of many of the papers, but nobody had collected all the problems. “This kind of single repository that anyone can access — labs realized that this could effectively be a benchmark.”
In January, a team of 24 researchers led by Google DeepMind shared a paper：https://arxiv.org/abs/2601.22401 solving four problems and finding old, forgotten solutions to nine more, after “using Gemini to systematically evaluate 700 conjectures labeled ‘Open’ in Bloom’s Erdős Problems database.” In May, a separate DeepMind team of 21 researchers announced：https://arxiv.org/abs/2605.22763 that “our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars.” (As of this article’s publication, Bloom’s database contains 565 solved problems and 652 open ones, but the DeepMind team narrowed their search to problems that have been written in formal logic.)
And, on May 20, OpenAI shared a solution：https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf to the unit distance problem, along with a blog post：https://openai.com/index/model-disproves-discrete-geometry-conjecture/ explaining the work and a companion paper：https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf that featured nine world-class mathematicians commenting on the correctness of the proof and the importance of what had been done (as well as presenting a streamlined human version of the result). Mathematicians had generally believed that Erdős’ conjecture — about how many evenly spaced points can be placed on a plane — was correct. To general surprise, OpenAI’s internal model found a counterexample. To do so, it had found a sophisticated way to use tools from an area of math called algebraic number theory. As Jacob Tsimerman：https://www.math.utoronto.ca/~jacobt/ of the University of Toronto wrote in the companion article, “This is a really impressive piece of work. … It is definitely an intimidating construction.”
In the same article, Tim Gowers：https://www.college-de-france.fr/en/chair/timothy-gowers-combinatorics-statutory-chair/biography of Cambridge and the Collège de France wrote that “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.”
The author on the paper that presented the original solution was given simply as “OpenAI.”
Terence Tao was 10 years old when he met Erdős.
Later, using techniques related to the ones the AI model had applied to the unit distance problem, a group of four mathematicians, including Bloom, disproved：https://arxiv.org/abs/2605.28781 a version of another long-standing Erdős conjecture. The “sum-product” conjecture：https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/ proposed that if you have sets of numbers, either their sum or their product must grow quickly. The mathematicians found a set of real numbers for which both the sum and the product grow more slowly than expected. The conjecture for integers remains open.
Figuring out what impact AI will have on math and mathematicians means not only looking to its most important results, but also examining how it changes the everyday practice of solving quotidian problems. Noga Alon, the Princeton mathematician, estimates that he has solved a few dozen Erdős problems over his career. He has now stopped trying. “Once AI started to solve them, there is no point anymore,” he said. Terry Tao has stepped away from the Erdős problem community to focus on getting work done.
Van Doorn, who for now still has his day job at a customer service company, said that LLMs “are clearly better at thinking and doing math than I am. I don’t hold a candle to current AI systems.” However, he added, “the eventual proofs that I write are simpler, more general, and easier to read for other people than the thing that ChatGPT came up with.” AI has indisputably boosted his productivity, and he’s still having fun. “If you want to play piano, you aren’t going to hire a piano-playing machine that does it better than you. You will play the piano because you like playing the piano. I enjoy thinking about numbers, doing math, writing papers. I’m not going to hire a paper-making machine that does it for me.”
For van Doorn, there is joy to be found in digesting the responses he gets from LLMs. “I’ve been doing a lot of math recently thanks to the Erdős-problems community. It used to be the case I just did everything all by myself, struggled alone in my room. I don’t know how it happened, but nowadays people contact me saying, ‘I have this idea. Do you want to join me thinking about this?’”
The increasing capability of AI has made it easier for people like Price or van Doorn, with less mathematical training, to solve puzzles, while making those puzzles less interesting to people like Alon who have devoted a lifetime to understanding them.
Nonetheless, “many and maybe most good mathematicians will use AI,” Alon said. He notes that a number of first-rate mathematicians have left academia to work at AI companies, not only because they are well paid to do so but because “maybe this is where the action now is.” In July 2026, on the same day that Tsimerman was awarded the Fields Medal：https://www.quantamagazine.org/jacob-tsimerman-wins-2026-fields-medal-for-andre-oort-conjecture-proof-20260723/, the highest honor in math, he announced that he was leaving academia for a job at OpenAI.
What Is the Unit Distance Problem? p]:my-6 [&>ul]:my-6 [&>ol]:my-6 [&>p]:text-3-5 [&>p]:leading-6.5 [&>li]:text-3-5 [&>li]:leading-6.5 [&_img.alignleft]:float-left [&_img.alignleft]:mr-5 [&_img.alignleft]:ml-0 [&_img.alignleft]:my-5 [&_img.alignright]:float-right [&_img.alignright]:ml-5 [&_img.alignright]:mr-0 [&_img.alignright]:my-5 [&_figure]:m-0 [&_figcaption]:relative [&_figcaption]:flex [&_figcaption]:flex-col [&_figcaption]:gap-2 [&_figcaption]:pt-2 [&_figcaption]:pb-4-5 [&_figcaption]:mt-0 [&_figcaption]:mb-6 &_figcaption]:font-pangram [&_figcaption]:after:content-[""] [&_figcaption]:after:absolute [&_figcaption]:after:bottom-0 [&_figcaption]:after:w-11 [&_figcaption]:after:h-0.5 [&_figcaption]:after:bg-gray-1a1 [&_.caption]:block [&_.caption]:font-pangram [&_.caption]:text-0xxs [&_.caption]:leading-4-5 [&_.caption]:m-0 [&_.attribution]:block [&_.attribution]:font-pangram [&_.attribution]:text-xs [&_.attribution]:leading-4-5 [&_.attribution]:m-0 [&_.attribution]:before:content-none" style="color: #000000;"> Say you want to arrange points in a plane to create pairs of points that are the same distance apart. How can you create the most pairs?
If you arrange your points so that they form polygons like squares and pentagons, you will get the same number of pairs as you have points.
But if you put points in a lattice, you can create substantially more equidistant pairs.
Erdős conjectured that it wasn’t possible to do substantially better than this. For 80 years, mathematicians generally believed he was correct.
But Erdős was mistaken. OpenAI found a pattern (similar to the one shown below) that, for a given number of points, produces more pairs than a lattice can.
扫码直达文章详情