{"@context":"https://schema.org","@type":"NewsArticle","generatedAt":"2026-08-11T09:21:12.743Z","headline":"为什么传奇的埃尔德什问题正被人工智能攻克","description":"OpenAI 于 2026 年 5 月 20 日宣布，其内部 AI 模型对埃尔德什 1946 年提出的\"单位距离\"问题给出了反例，成为首个由 AI 模型完成的历史性重要证明；8 月 1 日又宣布未发布模型 Astra 取得 10 项数学进展，其中解决了埃尔德什提出的另外 3 个问题。","url":"https://www.aioga.com/news/cmsgbawl0001iro5qajirh4qh/","mainEntityOfPage":"https://www.aioga.com/news/cmsgbawl0001iro5qajirh4qh/","datePublished":"2026-08-05T16:23:34.819Z","dateModified":"2026-08-05T16:23:34.819Z","inLanguage":"zh-CN","publisher":{"@type":"NewsMediaOrganization","name":"Aioga","url":"https://www.aioga.com"},"citation":["https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803","https://aihot.virxact.com/items/cmsgbawl0001iro5qajirh4qh"],"canonicalUrl":"https://www.aioga.com/news/cmsgbawl0001iro5qajirh4qh/","directAnswer":{"@type":"Answer","text":"据材料，OpenAI于2026年5月宣布，其内部模型为1946年提出的单位距离问题构造了反例；8月又称未发布模型Astra取得10项数学进展，其中涉及另外3个埃尔德什问题。文章将前者称为AI带来的首个具有历史意义的证明。","url":"https://www.aioga.com/news/cmsgbawl0001iro5qajirh4qh/","dateCreated":"2026-08-05T16:23:34.819Z","author":{"@type":"Organization","@id":"https://www.aioga.com/authors/aioga-editorial/#editorial-team","name":"Aioga Editorial Team","url":"https://www.aioga.com/authors/aioga-editorial/"}},"evidence":[{"@type":"CreativeWork","name":"quantamagazine.org source article","url":"https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803","datePublished":"2026-08-05T16:23:34.819Z","provider":{"@type":"Organization","name":"quantamagazine.org","url":"https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803"}},{"@type":"CreativeWork","name":"AIHot archive record","url":"https://aihot.virxact.com/items/cmsgbawl0001iro5qajirh4qh","datePublished":"2026-08-05T16:23:34.819Z","provider":{"@type":"Organization","name":"AIHot","url":"https://aihot.virxact.com/items/cmsgbawl0001iro5qajirh4qh"}}],"aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","originalPublisher":{"name":"quantamagazine.org","url":"https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803"},"geoDeepAnswer":null,"article":{"id":"cmsgbawl0001iro5qajirh4qh","slug":"cmsgbawl0001iro5qajirh4qh","url":"https://www.aioga.com/news/cmsgbawl0001iro5qajirh4qh/","title":"为什么传奇的埃尔德什问题正被人工智能攻克","title_en":"为什么埃尔德什问题正被人工智能攻克","summary":"OpenAI 于 2026 年 5 月 20 日宣布，其内部 AI 模型对埃尔德什 1946 年提出的\"单位距离\"问题给出了反例，成为首个由 AI 模型完成的历史性重要证明；8 月 1 日又宣布未发布模型 Astra 取得 10 项数学进展，其中解决了埃尔德什提出的另外 3 个问题。","source":"Hacker News 热门（buzzing.cc 中文翻译）","sourceUrl":"https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803","aiHotUrl":"https://aihot.virxact.com/items/cmsgbawl0001iro5qajirh4qh","publishedAt":"2026-08-05T16:23:34.819Z","category":"技巧观点","score":71,"selected":true,"articleBody":["Get the latest news delivered to your inbox.","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.","：https://www.quantamagazine.org/wp-content/uploads/2026/08/ErdosAI_Graphics_ErdosAI-Figure2-crMarkBelan-Desktopv1.svg","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."],"articleImages":[{"sourceUrl":"https://www.quantamagazine.org/wp-content/uploads/2022/10/Konstantin_Kakaes-160x160.jpg","alt":"","afterParagraph":3,"url":"/media/articles/cmsgbawl0001iro5qajirh4qh/5f1f67db5b4f63c3.jpg"},{"sourceUrl":"https://www.quantamagazine.org/wp-content/uploads/2026/08/Unit-distance-box-V2.svg","alt":"","afterParagraph":43,"url":"/media/articles/cmsgbawl0001iro5qajirh4qh/0926b71c8ae640c7.jpg"}],"mediaStatus":"ok","articleBodyZh":["将最新新闻直接发送到您的收件箱。","由西蒙基金会支持的编辑独立出版物。","通过点击您希望保存的文章旁边的“稍后阅读”图标来创建阅读列表。","输入搜索词并按回车","2026年5月20日，OpenAI发布了一则震动数学界的公告。一个内部的人工智能模型——一个公众无法获得的模型——提出了“单位距离”问题的反例：https://openai.com/index/model-disproves-discrete-geometry-conjecture/，该猜想由多产且游历广泛的匈牙利数学家保罗·埃尔德什于1946年提出。","埃尔德什提出了成千上万个问题，但这个问题很特别：它既容易解释，又在数学上深奥。这是历史上第一个来自人工智能模型的重要证明。虽然该模型的结果尚不确定——人类数学家将在几周内大幅改进它——它具有创新性，从数学的一个遥远分支引入了想法，而以前没有人成功地将其应用于该问题。而且它具有影响力：几天内，相关技术被用于解决其他重要问题。","然后在8月1日，OpenAI宣布：https://openai.com/index/ten-advances-in-mathematics/ 一个尚未发布的名为Astra的模型取得了10项额外的数学进展，包括解决埃尔德什提出的三道新问题。","许多数学家称这些发展是人工智能模型数学能力的一个相变。这些模型“正在显著改变数学研究的进行方式”，普林斯顿大学的Noga Alon：https://web.math.princeton.edu/~nalon/表示，他在数十年的职业生涯中解决了数十个埃尔德什问题。","保罗·埃尔德什，历史上最多产的数学家之一，在数学方面充满童趣，在权威面前却极为愤世嫉俗。","埃尔德什及他的猜想长期以来一直令数学家着迷。他不断地旅行——多年来几乎一直住在行李箱里，住在朋友家里，几乎一无所有。他在发表的论文和给世界各地数学家的信件中列出问题，经常附上奖金，这笔钱他会自掏腰包支付给第一个提出解决方案的人。奖励可能是象征性的10美元或25美元，或者，对于他认为重要或困难的问题，可能高达数千美元。埃尔德什于1996年在华沙参加数学会议时因心脏病去世，但一家总部位于爱荷华州的非营利基金会已经承诺：https://www.combinatoricsfoundation.org/erd%C5%91s-problems 来兑现他的奖金。","他是位备受喜爱的人物，但也非常古怪。他只穿丝绸，避免与他人接触。对权威深持怀疑态度的他，把大部分赚来的钱捐赠出去，并依靠朋友管理他的财务及其他实际事务。他把上帝称为“最高法西斯”，并靠持续摄入安非他命来维持他不断的数学创作。历史的讽刺之处在于，他提出的问题如今已成为世界上最大、最强大的科技公司验证实力的核心试炼场——实际上，也成为了一系列的公关胜利。","但很可能，如果不是一位名叫托马斯·布鲁姆的英国数学家：http://thomasbloom.org/，这一切都不会发生。","和埃尔德什一样，布鲁姆对数论和组合学都感兴趣。他的研究重点是一个被称为算术组合学的领域，它位于二者的交叉点。2014年获得博士学位后，布鲁姆在该领域崭露头角，获得了英国皇家学会的一个享有盛誉的奖学金：https://royalsociety.org/grants/university-research/，让他能够在几乎任何他想去的大学工作。（他现在在曼彻斯特大学。）","布鲁姆从记事起就很喜欢埃尔德什的风格。但他总是觉得很难跟踪哪些问题已经被解决，哪些则完全被遗忘。因此，在2023年初，他决定将尽可能多的问题汇集成一个列表。","他原本打算自己使用它。但他说：“我觉得如果我能随时随地访问它会更容易。”他想“干脆做一个网站，心里想着也许没人会用它。”他收集了几百个问题，并启动了 erdosproblems.com：http://erdosproblems.com。布鲁姆使用 ChatGPT 编写了运行该网站的 Python 代码，这在当时是一个大型语言模型能够做到的了不起的事情。使用它来直接协作数学研究仍然似乎只是一个遥远的可能性。","托马斯·布鲁姆的埃尔德什问题网站成为了数学最佳实践的聚集地。然后人工智能登场了。","他的目标不仅仅是完成清单上的项目。他想知道“现代数学，经常使用埃尔德什不知道的技巧，是否能够解决许多这些较为晦涩的问题，”他在博客文章中写道：https://www.erdosproblems.com/forum/thread/blog:1。 “我们最终将只剩下一些有趣且困难的问题，它们可以展示我们知识的极限。”","布鲁姆在策划问题列表方面做了关键工作：有时埃尔德什提出的问题表述模糊不清，而布鲁姆则弄清楚每个问题最合理的版本。 他不断向网站添加问题，渐渐地观众群也增长了。在2024年及2025年前八个月期间，列表上111个问题的状态从“未解决”改为“已解决”（尽管其中一些问题数年前就已解决，状态变化反映的是重新发现或验证了证明）。","然后，在2025年8月，一些同事建议布鲁姆添加评论功能，以便人们可以讨论他们感兴趣的问题。他能够很快实现这一功能，使用 ChatGPT 编写代码。到那时，他已经 catalog 了近1000个问题。","布鲁姆的时机很好。他使志同道合的人能够互相交流，他说，“这真的让一个社区逐渐形成。”大体上，评论是零散的——一个问题可能收到单条评论，指出一个例子或说明问题看起来有多难。但活动稳步增长，一些问题引发了陌生人之间深入的数学讨论。","“汤姆可能从未真正意识到这一点，但对我来说，这真的改变了我的人生，”布鲁姆网站上第四多产评论者沃特·范·多恩说：https：//www.erdosproblems.com/forum/user/Woett？all_posts=1。像许多2025年秋季开始活跃于该网站的人一样，van Doorn并不算是专业数学家。他说，他“为一家被其他公司雇佣做客户服务支持的公司工作”。但他也不算业余——十年前，他几乎完成了比利时鲁汶大学的数学硕士学位。2024年，部分受到他看到LLM日益强大的激励，他请了六个月假，专注于数学。当时，虽然他并不太想用AI，但他记得当时想：“现在我数学水平还是比AI强，但谁知道一年、两年、五年后会怎样？如果我想完成这些项目，并且想让它们属于我，现在就是时候。”","如果对于所有 n \\geq 1$，只有有限个 \\in A$ $latex $latex n + a$ 是平方自由的，则$latex称 A \\subseteq \\mathbb{N}$ 具有 P$ $latex$latex性质。如果存在无限多个 $latex n$，使得 $latex$latex n + a$ 对所有 < n$ $latex 都是平方自由的，则称 A$ 具有 Q$ $latex性质。具有 P$ 或 $latex Q$ 性质$latex$latex A = \\{a_1 < a_2 < \\cdots\\}$ 的序列必须多快地增加？","于是，2025年10月，van Doorn回到了他的本职工作，在问题1102的页面上留下了第一条评论：https：//www.erdosproblems.com/forum/thread/1102？order=oldest。埃尔德什于1981年提出的问题，探讨了“无平方”整数集合的性质——即没有重复质因数的整数。（例如，30是平方无的，因为它等于2×3×5;但18不是，因为它等于2×3×3;3重复。）","11月初，van Doorn在问题页面评论中分享了他未依赖AI就能找到答案的进展。","那天晚些时候，网站上的另一位评论者回复说，他发现了范·多恩论证中的一个漏洞。两人迅速交换了意见，范·多恩说服了他的对话者相信他的论证是正确的。“我现在明白你的论证是怎么回事了。太棒了！”另一位数学家回复道。那位数学家是陶哲轩（Terence Tao）：https://www.math.ucla.edu/~tao/，他是加利福尼亚大学洛杉矶分校的教授，可以说是当今最著名的数学家，同时也是无可争议的最有影响力的人物之一。（顺便提一下，当陶哲轩只有10岁的时候：https://blogs.ams.org/blogonmathblogs/2015/09/29/that-time-terrence-tao-won-500-from-paul-erdos/，他就与厄尔多什有过交集。）","布鲁姆的网站，看起来和感觉都像是过去的时代，正在成为互联网民主精神的一个典型例子。范·多恩说：“如果没有汤姆的网站和那里的评论区，这整个合作是不可能的。”无论你是否有教职，无论你是年轻还是年长，无论你在名校还是甚至是否在大学，你都可以在这里找到合作的伙伴，只要你有做数学的兴趣并且有好点子。","但是随着冬天的到来——大约在范·多恩发现自己正在与陶哲轩合作的同时——情况开始发生变化。","凯文·巴雷托（Kevin Barreto）和利亚姆·普莱斯（Liam Price）都二十出头，在2025年夏天在一个专注于人工智能的Discord服务器上成为朋友。巴雷托目前是剑桥大学的本科生；普莱斯在大学期间学习过一些数学，但未完成学业就离开了。到12月，他们确信最新的AI模型可能成功解决一些厄尔多什问题，于是开始向这些模型投送一批又一批的问题。他们很快意识到，如果告诉GPT-5.2某个问题的答案未知，它不会取得多大进展，所以正如巴雷托所说，他们学会了“以一种非常特别的方式提示它，让它误以为问题比实际更简单”。","设 $latex A \\subseteq \\mathbb{N}$ 是一个密度为零的集合。是否存在一个 $latex B$ 使得 $latex A \\subseteq B + B$ 并且对所有大的 $latex N$ 有 $latex |B \\cap \\{1, \\ldots, N\\}| = o(N^{1/2})$？","他们在埃尔德什问题333上取得了他们认为的首次胜利：https：//www.erdosproblems.com/forum/thread/333？order=oldest。圣诞节清晨，Barreto在Bloom的网站上发布了一份证明，写道：“据我们所知，我们相信这是首个完全自主解决Erdős问题的案例，此前人类未曾解决过。”尽管333涉及整数集合的和并不是特别重要的问题，但用AI解决它依然很重要。","但几小时后，另一位用户指出，埃尔德什本人在1977年发表的一篇论文中提出了333条的解决方案：https：//www.sciencedirect.com/science/article/pii/0022314X77900038。巴雷托承认了这个错误。他写道：“我正式请求网站所有成员，将当前标记为未解决的问题更加重视文献检索。”“作为一个已经两次上当的人，这真让人心如刀绞。”","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$latex a！b！\\mid n！（a + b – n）！$ $latex a + b > n + C \\log n$？","他并未气馁，他和普赖斯继续努力，到2026年1月4日，他们利用GPT-5.2 Pro找到了解决Erdős 728：https：//www.erdosproblems.com/forum/thread/728？order=oldest的问题，这是一个关于某些数字何时被其他数字整除的问题。这次没人能找到已有的证据。巴雷托使用另一款名为亚里士多德的人工智能工具（由一家名为Harmonic的初创公司开发）来证明证明逻辑上成立。软件工程师、论坛中唯一比Bloom、Tao和van Doorn更为高产的论坛参与者Nat Sothanaphan让ChatGPT写下了正式结果：https：//arxiv.org/abs/2601.07421 并发布到网上。","Price开发了一种方法论，用于指导如何让大语言模型（LLMs）解决开放性问题。首先，他会向聊天机器人请求一个解决方案。然后，他将该解决方案输入到一个新的聊天机器人实例中，要求它检查之前聊天机器人的工作。他会重复这个过程，直到得到看起来可行的解决方案。（这呼应了一些公司内部正在进行的工作，这些工作旨在创建所谓的“支架”或“框架”，自动化Price手动进行的迭代过程。）","Barreto和Price的论文只是过去几个月中AI至少部分解决的众多Erdős问题的一小部分。有多个原因使得这些问题特别成为LLMs的肥沃测试场。主要原因是，总体而言，Erdős问题多属于数论、组合数学和图论，这些数学领域被证明对大型语言模型比其他领域更易接近。这些问题在难度和数学重要性上也存在广泛差异。这种差异使它们适合用于测试能力同样差异较大的新兴技术。","Erdős的许多问题自创立之初就附带了金钱奖励，这是来自一位浪漫古怪人物的有趣激励。但现在，随着这些问题成为AI的非正式基准，其解决方案正在以“每个问题成本”的方式讨论——即解决这些问题所需的代币价格。","“我最近的许多论文大部分应归功于AI，”van Doorn说。“其中涉及的想法并非我自己提出的。”像很多在Erdős网站上活跃的人一样，van Doorn对LLMs让他能够更快速地完成更多事情感到兴奋。“如果我读到LLM提出的一个想法，我会消化它，尝试理解它，将其简化并推广，”他说。他利用AI更好地理解数学。","并非所有人都以此标准对待自己。布鲁姆说：“一个大问题是，人工智能被很多非数学家、没有深厚数学背景且无法验证输出的人使用。”“他们喜欢快速行动，让AI检查，它会越来越大。我们看到越来越多的100到200页的论文被发布。“我解决了这个定理;我让AI生成证明、检查证明并写论文。”但没有人类读过它，也没有人类会去读它。现在这成了巨大的挑战。”","对于任意$latex x$，$latex 一个 \\ 子集 [x， \\infty）$ 是一个本原整数集（因此 $latex A$ 中没有不同元素相互整除），那么是否$latex \\displaystyle\\sum_{a \\in A} \\frac{1}{a \\log a} < 1 + o（1），$ $latex o（1）$ 项 \\to 0$ $latex \\to 0$ $latex x \\到 \\infty$？","根据普赖斯自己的评估，他没有足够的数学理解来验证最终从大型语言模型中引出的解决方案。但在巴雷托的帮助下，他找到了足够有知识且愿意核查结果的数学家。普赖斯和巴雷托均与陶、斯坦福大学的贾里德·杜克·利希特曼（https：//mathematics.stanford.edu/people/jared-duker-lichtman）及其他杰出数学家共同撰写了2026年5月关于解决厄尔德什问题1196的论文：https：//arxiv.org/abs/2605.00301，这是他们较为重要的成果之一。（1196 询问所谓本原集的可能大小——即整数集合的集合，如 {2， 5， 9， 21}，其中没有任何数能整除其他数。）","Bloom 惊讶地发现，尽管 OpenAI、Google DeepMind 和几家初创公司给予了大量关注，但大多数新结果来自使用公开大型语言模型的业余爱好者和本科生，而非使用更先进内部模型的企业实验室。","但几周后，即2026年5月20日，OpenAI宣布他们解决了埃尔德什最著名的问题之一——单位距离问题，情况发生了变化。","在2026年的最初几个月，主要科技公司开始看到 erdosproblems.com 的机会。正如利希特曼解释的，“厄尔多什发表了超过1000篇论文。这些论文散落各处。” 匈牙利的一家研究所收集了许多论文的扫描图像，但没有人收集所有的问题。“这种任何人都可以访问的单一存储库——实验室意识到，这实际上可以成为一个基准。”","在一月份，由谷歌 DeepMind 领导的24名研究人员团队分享了一篇论文：https://arxiv.org/abs/2601.22401，解决了四个问题，并找到了另外九个被遗忘的旧解答，这是在“使用 Gemini 系统性评估 Bloom 的 Erdős Problems 数据库中标记为‘开放’的700个猜想”之后完成的。五月份，另一支由21名研究人员组成的 DeepMind 团队宣布：https://arxiv.org/abs/2605.22763，“我们最强大的智能体自主解决了353个开放 Erdős 问题中的9个，每个问题的成本只有几百美元。”（截至本文发表时，Bloom 的数据库包含565个已解决的问题和652个未解决的问题，但 DeepMind 团队将搜索范围缩小到已经用形式逻辑书写的问题。）","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。","范·多恩，目前仍在一家客户服务公司兼职，他表示大型语言模型（LLM）“在思考和做数学方面明显比我强。我完全比不上现有的人工智能系统。”然而，他补充道，“我最终写的证明，比ChatGPT想出来的东西更简单、更通用，也更容易被其他人阅读。”人工智能无可争议地提高了他的生产力，而他依然乐在其中。“如果你想弹钢琴，你不会去雇用一个比你弹得更好的钢琴机器。你会弹钢琴，因为你喜欢弹钢琴。我喜欢思考数字、做数学、写论文。我不会雇用一个帮我写论文的机器。”","对于范·多恩来说，从LLM得到的回应中获得乐趣是有意义的。“多亏了埃尔德什问题社区，我最近做了很多数学。以前我只是自己做所有事情，在房间里独自挣扎。我不知道是怎么发生的，但现在人们会联系我说，‘我有这个想法，你想和我一起思考吗？’”","人工智能能力的提升，使像普赖斯或范·多恩这样的数学训练较少的人更容易解决难题，同时也让这些难题对像艾龙这样一生致力于理解它们的人来说不那么有趣。","尽管如此，艾龙表示，“许多甚至可能大多数优秀数学家都会使用人工智能。”他注意到，一些一流数学家已经离开学术界去人工智能公司工作，不仅因为报酬丰厚，还因为“也许这才是现在的热点。”2026年7月，在齐默尔曼获得数学最高荣誉——菲尔兹奖的同一天：https://www.quantamagazine.org/jacob-tsimerman-wins-2026-fields-medal-for-andre-oort-conjecture-proof-20260723/，他宣布自己将离开学术界，去OpenAI工作。","单位距离问题是什么？ 假设你想在平面上排列点以创建相同距离的点对。如何才能创建最多的点对？","如果你排列的点形成像正方形和五边形这样的多边形，你将得到与点数相同数量的点对。","但如果你将点放入格子中，你可以创建更多的等距点对。","：https://www.quantamagazine.org/wp-content/uploads/2026/08/ErdosAI_Graphics_ErdosAI-Figure2-crMarkBelan-Desktopv1.svg","埃尔德什猜测不可能做得比这更好。在80年间，数学家们普遍认为他是正确的。","但埃尔德什错了。OpenAI发现了一个模式（类似于下面显示的），对于给定数量的点，产生的点对比格子中能得到的更多。"],"translationStatus":"translated","bodyOrigin":"source-page","editorial":{"summary":"据材料，OpenAI于2026年5月宣布，其内部模型为1946年提出的单位距离问题构造了反例；8月又称未发布模型Astra取得10项数学进展，其中涉及另外3个埃尔德什问题。文章将前者称为AI带来的首个具有历史意义的证明。","background":"埃尔德什提出过数千个数学问题，单位距离问题因表述简明且数学内涵深厚而受到重视。材料同时指出，该AI结果并非终局：人类数学家在数周内对其作出实质性改进，而模型的方法引入了此前未成功用于该问题的远支数学思路。","viewpoint":"Aioga 判断，这组进展的重点不只在于解出若干题目，还在于AI能否提出可被人类检验、改进和迁移的方法。值得关注的是，材料中的成果主要来自OpenAI公告及其未公开模型，外界对完整过程和可复现性的判断仍应以可核验材料为准。","implications":"材料援引数学家观点称，相关发展可能正在显著改变数学研究方式。Aioga 判断，若AI持续提供新构造、跨领域思路或可推进的中间结果，数学家的工作重心可能更多转向验证、推广与解释；但这不等同于AI已能独立完成所有高质量证明。","nextStep":"后续值得关注三点：单位距离反例及Astra所涉问题的完整证明是否公开；独立数学家能否复核并推广相关方法；以及后续成果是否能在公开模型或可重复的评测条件下得到验证。对“阶段性跃迁”的评价，宜结合这些证据持续更新。","evidenceRefs":["title","summary","articleBody","source"],"status":"published","aiGenerated":true,"autoApproved":true,"generatedBy":"aioga-editorial:gpt-5.6-sol","reviewedBy":"aioga-editorial-review:gpt-5.6-sol","generatedAt":"2026-08-05T17:43:44.962Z","sourceHash":"6cd639d4a6395caf","review":{"approved":true,"groundedness":95,"clarity":92,"duplicationRisk":10,"blockingIssues":[],"notes":["“8月”可改为“8月1日”，以与来源材料的具体日期保持一致。","“Aioga 判断”属于明确标注的编辑判断，未与来源事实混淆。"]},"validation":{"passed":true,"mode":"ai-auto","revisions":0,"checks":["schema","length","source-attribution","low-source-overlap","no-html","independent-ai-review"]}},"tags":["技巧观点","Hacker News 热门（buzzing.cc 中文翻译）"],"translations":{"zh-CN":{"title":"为什么传奇的埃尔德什问题正被人工智能攻克","summary":"OpenAI 于 2026 年 5 月 20 日宣布，其内部 AI 模型对埃尔德什 1946 年提出的\"单位距离\"问题给出了反例，成为首个由 AI 模型完成的历史性重要证明；8 月 1 日又宣布未发布模型 Astra 取得 10 项数学进展，其中解决了埃尔德什提出的另外 3 个问题。","category":"技巧观点","source":"quantamagazine.org","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"为什么传奇的埃尔德什问题正被人工智能攻克 - Aioga AI资讯","description":"OpenAI 于 2026 年 5 月 20 日宣布，其内部 AI 模型对埃尔德什 1946 年提出的\"单位距离\"问题给出了反例，成为首个由 AI 模型完成的历史性重要证明；8 月 1 日又宣布未发布模型 Astra 取得 10 项数学进展，其中解决了埃尔德什提出的另外 3 个问题。","url":"https://www.aioga.com/news/cmsgbawl0001iro5qajirh4qh/","articleBody":["将最新新闻直接发送到您的收件箱。","由西蒙基金会支持的编辑独立出版物。","通过点击您希望保存的文章旁边的“稍后阅读”图标来创建阅读列表。","输入搜索词并按回车","2026年5月20日，OpenAI发布了一则震动数学界的公告。一个内部的人工智能模型——一个公众无法获得的模型——提出了“单位距离”问题的反例：https://openai.com/index/model-disproves-discrete-geometry-conjecture/，该猜想由多产且游历广泛的匈牙利数学家保罗·埃尔德什于1946年提出。","埃尔德什提出了成千上万个问题，但这个问题很特别：它既容易解释，又在数学上深奥。这是历史上第一个来自人工智能模型的重要证明。虽然该模型的结果尚不确定——人类数学家将在几周内大幅改进它——它具有创新性，从数学的一个遥远分支引入了想法，而以前没有人成功地将其应用于该问题。而且它具有影响力：几天内，相关技术被用于解决其他重要问题。","然后在8月1日，OpenAI宣布：https://openai.com/index/ten-advances-in-mathematics/ 一个尚未发布的名为Astra的模型取得了10项额外的数学进展，包括解决埃尔德什提出的三道新问题。","许多数学家称这些发展是人工智能模型数学能力的一个相变。这些模型“正在显著改变数学研究的进行方式”，普林斯顿大学的Noga Alon：https://web.math.princeton.edu/~nalon/表示，他在数十年的职业生涯中解决了数十个埃尔德什问题。","保罗·埃尔德什，历史上最多产的数学家之一，在数学方面充满童趣，在权威面前却极为愤世嫉俗。","埃尔德什及他的猜想长期以来一直令数学家着迷。他不断地旅行——多年来几乎一直住在行李箱里，住在朋友家里，几乎一无所有。他在发表的论文和给世界各地数学家的信件中列出问题，经常附上奖金，这笔钱他会自掏腰包支付给第一个提出解决方案的人。奖励可能是象征性的10美元或25美元，或者，对于他认为重要或困难的问题，可能高达数千美元。埃尔德什于1996年在华沙参加数学会议时因心脏病去世，但一家总部位于爱荷华州的非营利基金会已经承诺：https://www.combinatoricsfoundation.org/erd%C5%91s-problems 来兑现他的奖金。","他是位备受喜爱的人物，但也非常古怪。他只穿丝绸，避免与他人接触。对权威深持怀疑态度的他，把大部分赚来的钱捐赠出去，并依靠朋友管理他的财务及其他实际事务。他把上帝称为“最高法西斯”，并靠持续摄入安非他命来维持他不断的数学创作。历史的讽刺之处在于，他提出的问题如今已成为世界上最大、最强大的科技公司验证实力的核心试炼场——实际上，也成为了一系列的公关胜利。","但很可能，如果不是一位名叫托马斯·布鲁姆的英国数学家：http://thomasbloom.org/，这一切都不会发生。","和埃尔德什一样，布鲁姆对数论和组合学都感兴趣。他的研究重点是一个被称为算术组合学的领域，它位于二者的交叉点。2014年获得博士学位后，布鲁姆在该领域崭露头角，获得了英国皇家学会的一个享有盛誉的奖学金：https://royalsociety.org/grants/university-research/，让他能够在几乎任何他想去的大学工作。（他现在在曼彻斯特大学。）","布鲁姆从记事起就很喜欢埃尔德什的风格。但他总是觉得很难跟踪哪些问题已经被解决，哪些则完全被遗忘。因此，在2023年初，他决定将尽可能多的问题汇集成一个列表。","他原本打算自己使用它。但他说：“我觉得如果我能随时随地访问它会更容易。”他想“干脆做一个网站，心里想着也许没人会用它。”他收集了几百个问题，并启动了 erdosproblems.com：http://erdosproblems.com。布鲁姆使用 ChatGPT 编写了运行该网站的 Python 代码，这在当时是一个大型语言模型能够做到的了不起的事情。使用它来直接协作数学研究仍然似乎只是一个遥远的可能性。","托马斯·布鲁姆的埃尔德什问题网站成为了数学最佳实践的聚集地。然后人工智能登场了。","他的目标不仅仅是完成清单上的项目。他想知道“现代数学，经常使用埃尔德什不知道的技巧，是否能够解决许多这些较为晦涩的问题，”他在博客文章中写道：https://www.erdosproblems.com/forum/thread/blog:1。 “我们最终将只剩下一些有趣且困难的问题，它们可以展示我们知识的极限。”","布鲁姆在策划问题列表方面做了关键工作：有时埃尔德什提出的问题表述模糊不清，而布鲁姆则弄清楚每个问题最合理的版本。 他不断向网站添加问题，渐渐地观众群也增长了。在2024年及2025年前八个月期间，列表上111个问题的状态从“未解决”改为“已解决”（尽管其中一些问题数年前就已解决，状态变化反映的是重新发现或验证了证明）。","然后，在2025年8月，一些同事建议布鲁姆添加评论功能，以便人们可以讨论他们感兴趣的问题。他能够很快实现这一功能，使用 ChatGPT 编写代码。到那时，他已经 catalog 了近1000个问题。","布鲁姆的时机很好。他使志同道合的人能够互相交流，他说，“这真的让一个社区逐渐形成。”大体上，评论是零散的——一个问题可能收到单条评论，指出一个例子或说明问题看起来有多难。但活动稳步增长，一些问题引发了陌生人之间深入的数学讨论。","“汤姆可能从未真正意识到这一点，但对我来说，这真的改变了我的人生，”布鲁姆网站上第四多产评论者沃特·范·多恩说：https：//www.erdosproblems.com/forum/user/Woett？all_posts=1。像许多2025年秋季开始活跃于该网站的人一样，van Doorn并不算是专业数学家。他说，他“为一家被其他公司雇佣做客户服务支持的公司工作”。但他也不算业余——十年前，他几乎完成了比利时鲁汶大学的数学硕士学位。2024年，部分受到他看到LLM日益强大的激励，他请了六个月假，专注于数学。当时，虽然他并不太想用AI，但他记得当时想：“现在我数学水平还是比AI强，但谁知道一年、两年、五年后会怎样？如果我想完成这些项目，并且想让它们属于我，现在就是时候。”","如果对于所有 n \\geq 1$，只有有限个 \\in A$ $latex $latex n + a$ 是平方自由的，则$latex称 A \\subseteq \\mathbb{N}$ 具有 P$ $latex$latex性质。如果存在无限多个 $latex n$，使得 $latex$latex n + a$ 对所有 < n$ $latex 都是平方自由的，则称 A$ 具有 Q$ $latex性质。具有 P$ 或 $latex Q$ 性质$latex$latex A = \\{a_1 < a_2 < \\cdots\\}$ 的序列必须多快地增加？","于是，2025年10月，van Doorn回到了他的本职工作，在问题1102的页面上留下了第一条评论：https：//www.erdosproblems.com/forum/thread/1102？order=oldest。埃尔德什于1981年提出的问题，探讨了“无平方”整数集合的性质——即没有重复质因数的整数。（例如，30是平方无的，因为它等于2×3×5;但18不是，因为它等于2×3×3;3重复。）","11月初，van Doorn在问题页面评论中分享了他未依赖AI就能找到答案的进展。","那天晚些时候，网站上的另一位评论者回复说，他发现了范·多恩论证中的一个漏洞。两人迅速交换了意见，范·多恩说服了他的对话者相信他的论证是正确的。“我现在明白你的论证是怎么回事了。太棒了！”另一位数学家回复道。那位数学家是陶哲轩（Terence Tao）：https://www.math.ucla.edu/~tao/，他是加利福尼亚大学洛杉矶分校的教授，可以说是当今最著名的数学家，同时也是无可争议的最有影响力的人物之一。（顺便提一下，当陶哲轩只有10岁的时候：https://blogs.ams.org/blogonmathblogs/2015/09/29/that-time-terrence-tao-won-500-from-paul-erdos/，他就与厄尔多什有过交集。）","布鲁姆的网站，看起来和感觉都像是过去的时代，正在成为互联网民主精神的一个典型例子。范·多恩说：“如果没有汤姆的网站和那里的评论区，这整个合作是不可能的。”无论你是否有教职，无论你是年轻还是年长，无论你在名校还是甚至是否在大学，你都可以在这里找到合作的伙伴，只要你有做数学的兴趣并且有好点子。","但是随着冬天的到来——大约在范·多恩发现自己正在与陶哲轩合作的同时——情况开始发生变化。","凯文·巴雷托（Kevin Barreto）和利亚姆·普莱斯（Liam Price）都二十出头，在2025年夏天在一个专注于人工智能的Discord服务器上成为朋友。巴雷托目前是剑桥大学的本科生；普莱斯在大学期间学习过一些数学，但未完成学业就离开了。到12月，他们确信最新的AI模型可能成功解决一些厄尔多什问题，于是开始向这些模型投送一批又一批的问题。他们很快意识到，如果告诉GPT-5.2某个问题的答案未知，它不会取得多大进展，所以正如巴雷托所说，他们学会了“以一种非常特别的方式提示它，让它误以为问题比实际更简单”。","设 $latex A \\subseteq \\mathbb{N}$ 是一个密度为零的集合。是否存在一个 $latex B$ 使得 $latex A \\subseteq B + B$ 并且对所有大的 $latex N$ 有 $latex |B \\cap \\{1, \\ldots, N\\}| = o(N^{1/2})$？","他们在埃尔德什问题333上取得了他们认为的首次胜利：https：//www.erdosproblems.com/forum/thread/333？order=oldest。圣诞节清晨，Barreto在Bloom的网站上发布了一份证明，写道：“据我们所知，我们相信这是首个完全自主解决Erdős问题的案例，此前人类未曾解决过。”尽管333涉及整数集合的和并不是特别重要的问题，但用AI解决它依然很重要。","但几小时后，另一位用户指出，埃尔德什本人在1977年发表的一篇论文中提出了333条的解决方案：https：//www.sciencedirect.com/science/article/pii/0022314X77900038。巴雷托承认了这个错误。他写道：“我正式请求网站所有成员，将当前标记为未解决的问题更加重视文献检索。”“作为一个已经两次上当的人，这真让人心如刀绞。”","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$latex a！b！\\mid n！（a + b – n）！$ $latex a + b > n + C \\log n$？","他并未气馁，他和普赖斯继续努力，到2026年1月4日，他们利用GPT-5.2 Pro找到了解决Erdős 728：https：//www.erdosproblems.com/forum/thread/728？order=oldest的问题，这是一个关于某些数字何时被其他数字整除的问题。这次没人能找到已有的证据。巴雷托使用另一款名为亚里士多德的人工智能工具（由一家名为Harmonic的初创公司开发）来证明证明逻辑上成立。软件工程师、论坛中唯一比Bloom、Tao和van Doorn更为高产的论坛参与者Nat Sothanaphan让ChatGPT写下了正式结果：https：//arxiv.org/abs/2601.07421 并发布到网上。","Price开发了一种方法论，用于指导如何让大语言模型（LLMs）解决开放性问题。首先，他会向聊天机器人请求一个解决方案。然后，他将该解决方案输入到一个新的聊天机器人实例中，要求它检查之前聊天机器人的工作。他会重复这个过程，直到得到看起来可行的解决方案。（这呼应了一些公司内部正在进行的工作，这些工作旨在创建所谓的“支架”或“框架”，自动化Price手动进行的迭代过程。）","Barreto和Price的论文只是过去几个月中AI至少部分解决的众多Erdős问题的一小部分。有多个原因使得这些问题特别成为LLMs的肥沃测试场。主要原因是，总体而言，Erdős问题多属于数论、组合数学和图论，这些数学领域被证明对大型语言模型比其他领域更易接近。这些问题在难度和数学重要性上也存在广泛差异。这种差异使它们适合用于测试能力同样差异较大的新兴技术。","Erdős的许多问题自创立之初就附带了金钱奖励，这是来自一位浪漫古怪人物的有趣激励。但现在，随着这些问题成为AI的非正式基准，其解决方案正在以“每个问题成本”的方式讨论——即解决这些问题所需的代币价格。","“我最近的许多论文大部分应归功于AI，”van Doorn说。“其中涉及的想法并非我自己提出的。”像很多在Erdős网站上活跃的人一样，van Doorn对LLMs让他能够更快速地完成更多事情感到兴奋。“如果我读到LLM提出的一个想法，我会消化它，尝试理解它，将其简化并推广，”他说。他利用AI更好地理解数学。","并非所有人都以此标准对待自己。布鲁姆说：“一个大问题是，人工智能被很多非数学家、没有深厚数学背景且无法验证输出的人使用。”“他们喜欢快速行动，让AI检查，它会越来越大。我们看到越来越多的100到200页的论文被发布。“我解决了这个定理;我让AI生成证明、检查证明并写论文。”但没有人类读过它，也没有人类会去读它。现在这成了巨大的挑战。”","对于任意$latex x$，$latex 一个 \\ 子集 [x， \\infty）$ 是一个本原整数集（因此 $latex A$ 中没有不同元素相互整除），那么是否$latex \\displaystyle\\sum_{a \\in A} \\frac{1}{a \\log a} < 1 + o（1），$ $latex o（1）$ 项 \\to 0$ $latex \\to 0$ $latex x \\到 \\infty$？","根据普赖斯自己的评估，他没有足够的数学理解来验证最终从大型语言模型中引出的解决方案。但在巴雷托的帮助下，他找到了足够有知识且愿意核查结果的数学家。普赖斯和巴雷托均与陶、斯坦福大学的贾里德·杜克·利希特曼（https：//mathematics.stanford.edu/people/jared-duker-lichtman）及其他杰出数学家共同撰写了2026年5月关于解决厄尔德什问题1196的论文：https：//arxiv.org/abs/2605.00301，这是他们较为重要的成果之一。（1196 询问所谓本原集的可能大小——即整数集合的集合，如 {2， 5， 9， 21}，其中没有任何数能整除其他数。）","Bloom 惊讶地发现，尽管 OpenAI、Google DeepMind 和几家初创公司给予了大量关注，但大多数新结果来自使用公开大型语言模型的业余爱好者和本科生，而非使用更先进内部模型的企业实验室。","但几周后，即2026年5月20日，OpenAI宣布他们解决了埃尔德什最著名的问题之一——单位距离问题，情况发生了变化。","在2026年的最初几个月，主要科技公司开始看到 erdosproblems.com 的机会。正如利希特曼解释的，“厄尔多什发表了超过1000篇论文。这些论文散落各处。” 匈牙利的一家研究所收集了许多论文的扫描图像，但没有人收集所有的问题。“这种任何人都可以访问的单一存储库——实验室意识到，这实际上可以成为一个基准。”","在一月份，由谷歌 DeepMind 领导的24名研究人员团队分享了一篇论文：https://arxiv.org/abs/2601.22401，解决了四个问题，并找到了另外九个被遗忘的旧解答，这是在“使用 Gemini 系统性评估 Bloom 的 Erdős Problems 数据库中标记为‘开放’的700个猜想”之后完成的。五月份，另一支由21名研究人员组成的 DeepMind 团队宣布：https://arxiv.org/abs/2605.22763，“我们最强大的智能体自主解决了353个开放 Erdős 问题中的9个，每个问题的成本只有几百美元。”（截至本文发表时，Bloom 的数据库包含565个已解决的问题和652个未解决的问题，但 DeepMind 团队将搜索范围缩小到已经用形式逻辑书写的问题。）","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。","范·多恩，目前仍在一家客户服务公司兼职，他表示大型语言模型（LLM）“在思考和做数学方面明显比我强。我完全比不上现有的人工智能系统。”然而，他补充道，“我最终写的证明，比ChatGPT想出来的东西更简单、更通用，也更容易被其他人阅读。”人工智能无可争议地提高了他的生产力，而他依然乐在其中。“如果你想弹钢琴，你不会去雇用一个比你弹得更好的钢琴机器。你会弹钢琴，因为你喜欢弹钢琴。我喜欢思考数字、做数学、写论文。我不会雇用一个帮我写论文的机器。”","对于范·多恩来说，从LLM得到的回应中获得乐趣是有意义的。“多亏了埃尔德什问题社区，我最近做了很多数学。以前我只是自己做所有事情，在房间里独自挣扎。我不知道是怎么发生的，但现在人们会联系我说，‘我有这个想法，你想和我一起思考吗？’”","人工智能能力的提升，使像普赖斯或范·多恩这样的数学训练较少的人更容易解决难题，同时也让这些难题对像艾龙这样一生致力于理解它们的人来说不那么有趣。","尽管如此，艾龙表示，“许多甚至可能大多数优秀数学家都会使用人工智能。”他注意到，一些一流数学家已经离开学术界去人工智能公司工作，不仅因为报酬丰厚，还因为“也许这才是现在的热点。”2026年7月，在齐默尔曼获得数学最高荣誉——菲尔兹奖的同一天：https://www.quantamagazine.org/jacob-tsimerman-wins-2026-fields-medal-for-andre-oort-conjecture-proof-20260723/，他宣布自己将离开学术界，去OpenAI工作。","单位距离问题是什么？ 假设你想在平面上排列点以创建相同距离的点对。如何才能创建最多的点对？","如果你排列的点形成像正方形和五边形这样的多边形，你将得到与点数相同数量的点对。","但如果你将点放入格子中，你可以创建更多的等距点对。","：https://www.quantamagazine.org/wp-content/uploads/2026/08/ErdosAI_Graphics_ErdosAI-Figure2-crMarkBelan-Desktopv1.svg","埃尔德什猜测不可能做得比这更好。在80年间，数学家们普遍认为他是正确的。","但埃尔德什错了。OpenAI发现了一个模式（类似于下面显示的），对于给定数量的点，产生的点对比格子中能得到的更多。"]},"en":{"title":"Why the legendary Erdős problem is being tackled by artificial intelligence","summary":"On May 20, 2026, OpenAI announced that its internal AI model provided a counterexample to the 'unit distance' problem proposed by Erdős in 1946, becoming the first historic proof completed by an AI model; on August 1, it also announced that its unreleased model Astra had achieved 10 mathematical advancements, including solving three other problems proposed by Erdős.","category":"Insights","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Why the legendary Erdős problem is being tackled by artificial intelligence - Aioga AI News","description":"On May 20, 2026, OpenAI announced that its internal AI model provided a counterexample to the 'unit distance' problem proposed by Erdős in 1946, becoming the first historic proof c...","url":"https://www.aioga.com/en/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:02:35.173Z"},"ja":{"title":"なぜ伝説のエルデシュ問題が人工知能によって解かれつつあるのか","summary":"OpenAIは2026年5月20日に、社内のAIモデルがエルデシュが1946年に提起した「単位距離」問題に対して反例を示し、AIモデルによる初の歴史的に重要な証明となったことを発表した。8月1日には、未公開モデルAstraが10の数学的進展を達成し、その中でエルデシュが提起した他の3つの問題を解決したことを発表した。","category":"ヒントと視点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"なぜ伝説のエルデシュ問題が人工知能によって解かれつつあるのか - Aioga AIニュース","description":"OpenAIは2026年5月20日に、社内のAIモデルがエルデシュが1946年に提起した「単位距離」問題に対して反例を示し、AIモデルによる初の歴史的に重要な証明となったことを発表した。8月1日には、未公開モデルAstraが10の数学的進展を達成し、その中でエルデシュが提起した他の3つの問題を解決したことを発表した。","url":"https://www.aioga.com/ja/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:02:36.120Z"},"ko":{"title":"왜 전설적인 엘더슈 문제를 인공지능이 해결하고 있는가","summary":"OpenAI는 2026년 5월 20일, 자사의 내부 AI 모델이 엘데시가 1946년에 제기한 '단위 거리' 문제에 대한 반례를 제시하여 AI 모델에 의해 이루어진 최초의 역사적으로 중요한 증명이 되었다고 발표했으며, 8월 1일에는 공개되지 않은 모델 Astra가 수학 분야에서 10개의 진전을 이루었고 그 중 3개의 문제는 엘데시가 제기한 다른 문제들을 해결했다고 발표했다.","category":"인사이트","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"왜 전설적인 엘더슈 문제를 인공지능이 해결하고 있는가 - Aioga AI 뉴스","description":"OpenAI는 2026년 5월 20일, 자사의 내부 AI 모델이 엘데시가 1946년에 제기한 '단위 거리' 문제에 대한 반례를 제시하여 AI 모델에 의해 이루어진 최초의 역사적으로 중요한 증명이 되었다고 발표했으며, 8월 1일에는 공개되지 않은 모델 Astra가 수학 분야에서 10개의 진전을 이루었고 그 중 3개의 문제는...","url":"https://www.aioga.com/ko/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:03:23.636Z"},"es":{"title":"Por qué el legendario problema de Erdős está siendo resuelto por la inteligencia artificial","summary":"OpenAI anunció el 20 de mayo de 2026 que su modelo de IA interno proporcionó un contraejemplo al problema de \"distancia unitaria\" planteado por Erdős en 1946, convirtiéndose en la primera demostración histórica importante realizada por un modelo de IA; el 1 de agosto anunció que el modelo no publicado Astra logró 10 avances matemáticos, resolviendo además 3 problemas más planteados por Erdős.","category":"Ideas","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Por qué el legendario problema de Erdős está siendo resuelto por la inteligencia artificial - Aioga Noticias de IA","description":"OpenAI anunció el 20 de mayo de 2026 que su modelo de IA interno proporcionó un contraejemplo al problema de \"distancia unitaria\" planteado por Erdős en 1946, convirtiéndose en la...","url":"https://www.aioga.com/es/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:03:19.748Z"},"fr":{"title":"Pourquoi le légendaire problème d’Erdős est en train d’être résolu par l’intelligence artificielle","summary":"OpenAI a annoncé le 20 mai 2026 que son modèle d'IA interne avait fourni un contre-exemple au problème de \"distance unité\" proposé par Erdős en 1946, devenant ainsi la première démonstration historiquement importante réalisée par un modèle d'IA ; le 1er août, elle a annoncé que son modèle non publié Astra avait réalisé 10 avancées mathématiques, dont la résolution de 3 autres problèmes proposés par Erdős.","category":"Analyses","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Pourquoi le légendaire problème d’Erdős est en train d’être résolu par l’intelligence artificielle - Aioga Actualités IA","description":"OpenAI a annoncé le 20 mai 2026 que son modèle d'IA interne avait fourni un contre-exemple au problème de \"distance unité\" proposé par Erdős en 1946, devenant ainsi la première dém...","url":"https://www.aioga.com/fr/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:04:06.231Z"},"de":{"title":"Warum das legendäre Erdős-Problem gerade von künstlicher Intelligenz gelöst wird","summary":"OpenAI kündigte am 20. Mai 2026 an, dass sein internes KI-Modell ein Gegenbeispiel für das von Erdős 1946 gestellte \"Einheitsabstands\"-Problem geliefert habe und damit der erste historisch bedeutende Beweis sei, der von einem KI-Modell erbracht wurde; am 1. August wurde außerdem angekündigt, dass das nicht veröffentlichte Modell Astra 10 mathematische Fortschritte erzielt habe, darunter die Lösung von drei weiteren von Erdős gestellten Problemen.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Warum das legendäre Erdős-Problem gerade von künstlicher Intelligenz gelöst wird - Aioga KI-News","description":"OpenAI kündigte am 20. Mai 2026 an, dass sein internes KI-Modell ein Gegenbeispiel für das von Erdős 1946 gestellte \"Einheitsabstands\"-Problem geliefert habe und damit der erste hi...","url":"https://www.aioga.com/de/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:04:10.023Z"},"pt-BR":{"title":"Por que o lendário problema de Erdős está sendo resolvido pela inteligência artificial","summary":"A OpenAI anunciou em 20 de maio de 2026 que seu modelo interno de IA forneceu um contraexemplo para o problema da \"distância unitária\" proposto por Erdős em 1946, tornando-se a primeira prova histórica significativa realizada por um modelo de IA; em 1º de agosto, anunciou que o modelo não divulgado Astra alcançou 10 avanços matemáticos, incluindo a solução de outros 3 problemas propostos por Erdős.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Por que o lendário problema de Erdős está sendo resolvido pela inteligência artificial - Aioga Notícias de IA","description":"A OpenAI anunciou em 20 de maio de 2026 que seu modelo interno de IA forneceu um contraexemplo para o problema da \"distância unitária\" proposto por Erdős em 1946, tornando-se a pri...","url":"https://www.aioga.com/pt-BR/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:04:43.448Z"},"ru":{"title":"Почему легендарная проблема Эрдеша решается с помощью искусственного интеллекта","summary":"OpenAI 20 мая 2026 года объявила, что её внутренная AI-модель предоставила контрпример к проблеме «единичного расстояния», предложенной Эрдешем в 1946 году, став первой исторически значимой теоремой, выполненной AI-моделью; 1 августа было объявлено, что непризнанная модель Astra достигла 10 математических достижений, включая решение ещё 3 проблем, предложенных Эрдешем.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Почему легендарная проблема Эрдеша решается с помощью искусственного интеллекта - Aioga Новости ИИ","description":"OpenAI 20 мая 2026 года объявила, что её внутренная AI-модель предоставила контрпример к проблеме «единичного расстояния», предложенной Эрдешем в 1946 году, став первой исторически...","url":"https://www.aioga.com/ru/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:04:54.974Z"},"ar":{"title":"لماذا يتمكن الذكاء الاصطناعي من حل مسألة إيرديش الأسطورية","summary":"أعلنت شركة OpenAI في 20 مايو 2026 أن نموذج الذكاء الاصطناعي الداخلي الخاص بها قدم أمثلة مضادة لمشكلة \"المسافة الواحدة\" التي قدمها إيرديش عام 1946، ليصبح أول إثبات تاريخي مهم يتم بواسطة نموذج ذكاء اصطناعي؛ وفي 1 أغسطس أعلنت أيضًا أن النموذج غير المنشور Astra أحرز 10 تقدمات رياضية، بما في ذلك حل 3 مشاكل أخرى طرحها إيرديش.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"لماذا يتمكن الذكاء الاصطناعي من حل مسألة إيرديش الأسطورية - Aioga أخبار الذكاء الاصطناعي","description":"أعلنت شركة OpenAI في 20 مايو 2026 أن نموذج الذكاء الاصطناعي الداخلي الخاص بها قدم أمثلة مضادة لمشكلة \"المسافة الواحدة\" التي قدمها إيرديش عام 1946، ليصبح أول إثبات تاريخي مهم يتم بو...","url":"https://www.aioga.com/ar/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:05:44.017Z"},"hi":{"title":"क्यों लंबित एल्डरश समस्या को कृत्रिम बुद्धिमत्ता द्वारा हल किया जा रहा है","summary":"OpenAI ने 20 मई 2026 को घोषणा की कि उसके आंतरिक AI मॉडल ने एर्ल्ड्श द्वारा 1946 में प्रस्तुत \"एकक दूरी\" समस्या का विरोधाभास उदाहरण प्रस्तुत किया, और यह AI मॉडल द्वारा पूरा किया गया पहला ऐतिहासिक महत्वपूर्ण प्रमाण बन गया; 1 अगस्त को उसने यह भी घोषणा की कि अनजारी मॉडल Astra ने 10 गणितीय प्रगति हासिल की हैं, जिनमें एर्ल्ड्श द्वारा प्रस्तुत अन्य 3 समस्याओं का समाधान भी शामिल है।","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"क्यों लंबित एल्डरश समस्या को कृत्रिम बुद्धिमत्ता द्वारा हल किया जा रहा है - Aioga AI समाचार","description":"OpenAI ने 20 मई 2026 को घोषणा की कि उसके आंतरिक AI मॉडल ने एर्ल्ड्श द्वारा 1946 में प्रस्तुत \"एकक दूरी\" समस्या का विरोधाभास उदाहरण प्रस्तुत किया, और यह AI मॉडल द्वारा पूरा किया गया...","url":"https://www.aioga.com/hi/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:05:39.250Z"},"it":{"title":"Perché il leggendario problema di Erdős viene risolto dall'intelligenza artificiale","summary":"OpenAI ha annunciato il 20 maggio 2026 che il suo modello AI interno ha fornito un controesempio al problema della \"distanza unitaria\" proposto da Erdős nel 1946, diventando la prima dimostrazione storicamente significativa completata da un modello AI; il 1º agosto ha poi annunciato che il modello non pubblicato Astra ha raggiunto 10 progressi matematici, risolvendo tra l'altro altri 3 problemi proposti da Erdős.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Perché il leggendario problema di Erdős viene risolto dall'intelligenza artificiale - Aioga Notizie IA","description":"OpenAI ha annunciato il 20 maggio 2026 che il suo modello AI interno ha fornito un controesempio al problema della \"distanza unitaria\" proposto da Erdős nel 1946, diventando la pri...","url":"https://www.aioga.com/it/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:06:32.926Z"},"nl":{"title":"Waarom het legendarische Erdős-probleem door kunstmatige intelligentie wordt opgelost","summary":"OpenAI kondigde op 20 mei 2026 aan dat hun interne AI-model een tegenvoorbeeld had geleverd voor het \"unit distance\"-probleem dat Erdős in 1946 had voorgesteld, waarmee het het eerste historisch belangrijke bewijs door een AI-model werd geleverd; op 1 augustus werd bovendien aangekondigd dat het niet-uitgebrachte model Astra 10 wiskundige doorbraken had bereikt, waaronder de oplossing van nog 3 door Erdős voorgestelde problemen.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Waarom het legendarische Erdős-probleem door kunstmatige intelligentie wordt opgelost - Aioga AI-nieuws","description":"OpenAI kondigde op 20 mei 2026 aan dat hun interne AI-model een tegenvoorbeeld had geleverd voor het \"unit distance\"-probleem dat Erdős in 1946 had voorgesteld, waarmee het het eer...","url":"https://www.aioga.com/nl/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:06:23.397Z"},"tr":{"title":"Neden efsanevi Erdős problemi yapay zeka tarafından çözülüyor","summary":"OpenAI, 20 Mayıs 2026'da, iç AI modelinin Erdős'ün 1946'da ortaya attığı \"birim mesafe\" sorununa karşı örnek sunduğunu ve bu alanda AI modeli tarafından gerçekleştirilen ilk tarihî öneme sahip kanıt haline geldiğini açıkladı; 1 Ağustos'ta ise yayımlanmayan Astra modelinin 10 matematiksel ilerleme kaydettiğini ve bu ilerlemeler arasında Erdős tarafından ortaya atılan diğer 3 sorunun çözüldüğünü duyurdu.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Neden efsanevi Erdős problemi yapay zeka tarafından çözülüyor - Aioga AI Haberleri","description":"OpenAI, 20 Mayıs 2026'da, iç AI modelinin Erdős'ün 1946'da ortaya attığı \"birim mesafe\" sorununa karşı örnek sunduğunu ve bu alanda AI modeli tarafından gerçekleştirilen ilk tarihî...","url":"https://www.aioga.com/tr/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:07:15.540Z"},"vi":{"title":"Tại sao vấn đề Erdős huyền thoại đang bị trí tuệ nhân tạo giải quyết","summary":"OpenAI vào ngày 20 tháng 5 năm 2026 đã công bố rằng mô hình AI nội bộ của họ đã đưa ra phản ví dụ cho vấn đề \"khoảng cách đơn vị\" do Erdős đề xuất năm 1946, trở thành minh chứng quan trọng lịch sử đầu tiên được hoàn thành bởi mô hình AI; vào ngày 1 tháng 8, họ lại công bố rằng mô hình chưa phát hành Astra đã đạt được 10 tiến bộ toán học, trong đó giải quyết thêm 3 vấn đề khác do Erdős đề xuất.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Tại sao vấn đề Erdős huyền thoại đang bị trí tuệ nhân tạo giải quyết - Tin tức AI Aioga","description":"OpenAI vào ngày 20 tháng 5 năm 2026 đã công bố rằng mô hình AI nội bộ của họ đã đưa ra phản ví dụ cho vấn đề \"khoảng cách đơn vị\" do Erdős đề xuất năm 1946, trở thành minh chứng qu...","url":"https://www.aioga.com/vi/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:07:12.247Z"},"id":{"title":"Mengapa masalah Erdős yang legendaris sedang dipecahkan oleh kecerdasan buatan","summary":"OpenAI pada 20 Mei 2026 mengumumkan bahwa model AI internalnya memberikan kontra contoh untuk masalah 'jarak satuan' yang diajukan oleh Erdős pada tahun 1946, menjadi bukti penting bersejarah pertama yang diselesaikan oleh model AI; pada 1 Agustus juga diumumkan bahwa model yang belum dirilis, Astra, mencapai 10 kemajuan dalam matematika, termasuk penyelesaian 3 masalah lain yang diajukan oleh Erdős.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Mengapa masalah Erdős yang legendaris sedang dipecahkan oleh kecerdasan buatan - Berita AI Aioga","description":"OpenAI pada 20 Mei 2026 mengumumkan bahwa model AI internalnya memberikan kontra contoh untuk masalah 'jarak satuan' yang diajukan oleh Erdős pada tahun 1946, menjadi bukti penting...","url":"https://www.aioga.com/id/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:08:00.880Z"},"th":{"title":"ทำไมปัญหา Elders ของตำนานถึงกำลังถูกเอไอทำลาย","summary":"OpenAI ประกาศเมื่อวันที่ 20 พฤษภาคม 2026 ว่าโมเดล AI ภายในของตนได้ให้ตัวอย่างที่ขัดแย้งกับปัญหา \"ระยะทางหน่วย\" ที่ Erdős เสนอในปี 1946 กลายเป็นการพิสูจน์สำคัญทางประวัติศาสตร์ครั้งแรกที่ทำโดยโมเดล AI; ในวันที่ 1 สิงหาคม ประกาศว่าโมเดลที่ยังไม่เปิดตัว Astra ได้ทำความก้าวหน้าทางคณิตศาสตร์ 10 เรื่อง ซึ่งรวมถึงการแก้ปัญหาอีก 3 ข้อที่ Erdős เสนอ","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ทำไมปัญหา Elders ของตำนานถึงกำลังถูกเอไอทำลาย - ข่าว AI Aioga","description":"OpenAI ประกาศเมื่อวันที่ 20 พฤษภาคม 2026 ว่าโมเดล AI ภายในของตนได้ให้ตัวอย่างที่ขัดแย้งกับปัญหา \"ระยะทางหน่วย\" ที่ Erdős เสนอในปี 1946 กลายเป็นการพิสูจน์สำคัญทางประวัติศาสตร์ครั้งแ...","url":"https://www.aioga.com/th/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:08:03.318Z"},"pl":{"title":"Dlaczego legendarny problem Erdősa jest rozwiązywany przez sztuczną inteligencję","summary":"OpenAI ogłosiło 20 maja 2026 roku, że jego wewnętrzny model AI dostarczył kontrprzykład do problemu 'jednostkowej odległości' zaproponowanego przez Erdősa w 1946 roku, stając się pierwszym historycznie ważnym dowodem wykonanym przez model AI; 1 sierpnia ogłoszono ponadto, że nieopublikowany model Astra osiągnął 10 postępów w matematyce, w tym rozwiązanie trzech innych problemów zaproponowanych przez Erdősa.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"Dlaczego legendarny problem Erdősa jest rozwiązywany przez sztuczną inteligencję - Aioga Wiadomości AI","description":"OpenAI ogłosiło 20 maja 2026 roku, że jego wewnętrzny model AI dostarczył kontrprzykład do problemu 'jednostkowej odległości' zaproponowanego przez Erdősa w 1946 roku, stając się p...","url":"https://www.aioga.com/pl/news/cmsgbawl0001iro5qajirh4qh/","contentTranslated":true,"sourceHash":"f4407188809bacee","translatedAt":"2026-08-05T17:08:49.821Z"}}}}