Why the Legendary Erdős Problems Are Falling to AI
O n May 20, 2026, OpenAI made an announcement that shook the mathematical world.
2026 年 5 月 20 日，OpenAI 发布了震惊数学界的公告。
An internal AI model — one not available to the public — had come up with a counterexample to the “unit distance” problem , a conjecture made in 1946 by Paul Erdős, the prolific, itinerant Hungarian mathematician.
一个内部人工智能模型——一个不向公众开放的模型——提出了“单位距离”问题的反例，“单位距离”问题是多产的匈牙利数学家 Paul Erdős 在 1946 年提出的猜想。
Erdős posed thousands of questions, but this one was special: It was both simple to explain and mathematically deep.
埃尔多斯提出了数千个问题，但这个问题很特别：它既易于解释，又具有数学深度。
It was the first historically significant proof to come from an AI model.
这是来自人工智能模型的第一个具有历史意义的证据。
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.
尽管该模型的结果不是确定的——人类数学家将在几周内对其进行大幅改进——但它具有创新性，引入了一个遥远的数学分支的想法，而以前没有人成功地将其应用于这个问题。
And it was influential: Within a few days, related techniques were used to solve other important problems.
它具有影响力：几天之内，相关技术就被用来解决其他重要问题。
Then on August 1, OpenAI announced that an unreleased model named Astra made 10 additional mathematical advances, including finding solutions to three more problems posed by Erdős.
然后在 8 月 1 日，OpenAI 宣布一个名为 Astra 的未发布模型取得了 10 项额外的数学进展，包括找到 Erdős 提出的另外三个问题的解决方案。
Many mathematicians have hailed developments such as these as a phase transition in the mathematical capability of AI models.
许多数学家称赞此类进展是人工智能模型数学能力的阶段性转变。
These models are “changing dramatically the way mathematical research is being done,” said Noga Alon of Princeton University, who has solved dozens of Erdős problems over his decades-long career.
普林斯顿大学的诺加·阿隆 (Noga Alon) 表示，这些模型“正在极大地改变数学研究的方式”。他在数十年的职业生涯中解决了数十个埃尔多斯问题。
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.
保罗·埃尔多斯（Paul Erdős）是历史上最多产的数学家之一，他在数学方面非常异想天开，在谈到权威时又非常愤世嫉俗。
Photo by George Csicsery from the documentary N is a Number: A Portrait of Paul Erdős ©1993.
摄影：George Csicsery，摘自纪录片《N 是一个数字：Paul Erdős 的肖像》©1993。
All Rights Reserved.
版权所有。
Erdős and his conjectures have long fascinated mathematicians.
埃尔多斯和他的猜想长期以来一直令数学家着迷。
He traveled constantly — living out of a suitcase for years at a time, staying with friends, owning almost nothing.
他经常旅行——一次好几年都提着行李箱生活，和朋友住在一起，几乎一无所有。
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.
他在发表的论文和写给世界各地数学家的信件中滔滔不绝地讲述问题，并经常附上奖金，他会自掏腰包支付给第一个提出解决方案的人。
The reward might be a token $10 or $25, or, for problems he considered important or difficult, it could range into the thousands.
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，奖励可能高达数千美元。
Erdős died of a heart attack in 1996 while attending a math conference in Warsaw, but a nonprofit foundation based in Iowa has promised to make good on his bounties.
1996 年，埃尔多斯在华沙参加数学会议时因心脏病去世，但爱荷华州的一个非营利基金会已承诺兑现对他的赏金。
He was a beloved figure, but also a downright weird one.
他是一个受人喜爱的人物，但也是一个彻头彻尾的怪人。
He only wore silk, and he avoided the touch of other people.
他只穿丝绸，避免别人的接触。
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.
他对权威非常愤世嫉俗，他捐出了大部分赚来的钱，并依靠朋友来管理他的财务和其他实际事务。
He referred to God as the “Supreme Fascist” and fueled his incessant output of mathematical ideas with a steady diet of amphetamines.
他将上帝称为“最高法西斯分子”，并通过稳定的安非他明饮食来推动他不断产出的数学思想。
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.
他提出的问题现在已成为世界上最大、最强大的科技公司的中心试验场，实际上是一系列公关政变，这是历史的一个奇怪的讽刺。
But in all likelihood none of this would have happened had it not been for an English mathematician named Thomas Bloom .
但如果没有一位名叫托马斯·布鲁姆的英国数学家，这一切很可能都不会发生。
Like Erdős, Bloom was interested in both number theory and combinatorics.
和埃尔多斯一样，布鲁姆对数论和组合学都感兴趣。
His focus has been an area called arithmetic combinatorics, which lies at the intersection of the two.
他的重点是算术组合学领域，该领域位于两者的交叉点。
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 , which let him work at almost any university he wanted to.
2014 年获得博士学位后，布鲁姆成为该领域的后起之秀，获得了英国皇家学会的著名奖学金，这让他可以在几乎任何他想去的大学工作。
(He’s now at the University of Manchester.)
（他现在在曼彻斯特大学。）
Bloom has liked Erdős’ style for as long as he can remember.
布鲁姆从记事起就喜欢埃尔多斯的风格。
But he always found it hard to keep track of which problems had been solved and which had been forgotten entirely.
但他总是发现很难追踪哪些问题已经解决，哪些问题被完全遗忘。
So in early 2023, he decided to gather as many problems as he could into a list.
因此，在 2023 年初，他决定将尽可能多的问题收集到一个清单中。
He intended it for his own use.
他打算将其供自己使用。
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 .
但“我认为如果我无论身在何处都可以访问它会更容易，”他说；他认为他“不妨制作一个网站，有点期望也许没有人会使用它。”他收集了几百个问题并推出了 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.
Bloom 使用 ChatGPT 编写了运行该网站的 Python 代码，这在当时对于大型语言模型来说是一件了不起的事情。
Using one to collaborate on the math itself still seemed like only a distant possibility.
使用一个人来合作研究数学本身似乎仍然是一种遥远的可能性。
Thomas Bloom’s website of Erdős problems became a home for mathematics at its best.
托马斯·布鲁姆 (Thomas Bloom) 的埃尔多斯问题网站成为数学最好的家园。
Then AI came on the scene.
然后AI登场了。
His goal was not just to cross items off a list.
他的目标不仅仅是删除清单上的项目。
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 .
他在一篇博客文章中写道，他想知道“现代数学通常使用埃尔多斯未知的技术，是否可以解决许多这些更晦涩难懂的问题”。
“We will then be left with a core of interesting, difficult problems, which can serve to demonstrate the limits of our knowledge.”
“然后我们将面临一些有趣而困难的核心问题，这些问题可以证明我们知识的局限性。”
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.
布鲁姆在整理这份清单方面做了至关重要的工作：有时，埃尔多斯以模棱两可或不明确的方式陈述问题，布鲁姆找出每个问题最合理的版本应该是什么。
He kept adding problems to the site, and gradually its audience grew.
他不断向网站添加问题，网站的受众逐渐增加。
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).
在 2024 年和 2025 年前 8 个月中，列表中 111 个问题的状态从“开放”更改为“已解决”（尽管其中一些问题早在几年前就已解决，其状态变化反映了证明的重新发现或验证）。
Then, in August 2025, some colleagues suggested that Bloom add a commenting function, so that people could talk about problems they were interested in.
然后，在 2025 年 8 月，一些同事建议 Bloom 添加评论功能，以便人们可以谈论他们感兴趣的问题。
He was able to do so quickly, using ChatGPT to write the code.
他使用 ChatGPT 编写代码，很快就做到了这一点。
By now he’d cataloged nearly 1,000 problems.
到目前为止，他已经列出了近 1,000 个问题。
Bloom’s timing was good.
布鲁姆的时机恰到好处。
He made it possible for like-minded people to talk to one another, and that “really let a community build up,” he said.
他让志趣相投的人能够互相交谈，这“真正让社区得以建立，”他说。
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.
大多数情况下，评论都是零星的——一个问题可能会吸引一条评论，指出一个例子或指出问题看起来有多难。
But activity steadily grew, and some problems catalyzed nuanced mathematical discussions between strangers.
但活动稳步增长，一些问题引发了陌生人之间细致入微的数学讨论。
“Tom probably never really realized this, but for me it’s honestly changed my life,” said Wouter van Doorn, the fourth-most-prolific commenter on Bloom’s website.
“汤姆可能从来没有真正意识到这一点，但对我来说，这确实改变了我的生活，”布鲁姆网站上第四多产的评论员沃特·范·多恩 (Wouter van Doorn) 说。
Like many people who became active on the site in the autumn of 2025, van Doorn isn’t exactly a professional mathematician.
与 2025 年秋季活跃在该网站上的许多人一样，范·多恩 (van Doorn) 并不完全是一名专业数学家。
He works “for a company that gets hired by other companies to do customer service support,” as he put it.
正如他所说，他“为一家被其他公司雇佣来提供客户服务支持的公司工作”。
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.
但他也不完全是一个业余爱好者——十年前，他几乎在比利时鲁汶大学完成了数学硕士学位。
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.
2024 年，由于看到法学硕士的能力越来越强，他请了六个月的假来专注于数学。
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?
当时，虽然他并不是特别想使用人工智能，但他记得自己在想：“现在我的数学水平仍然比人工智能更好，但谁知道一年、两年、五年后会怎样呢？
If I want to finish these projects, and I want them to be mine, now is the time.”
如果我想完成这些项目，并且希望它们属于我，那么现在就是时候了。”
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.
我们说 $latex A \subseteq \mathbb{N}$ 具有属性 $latex P$ 如果对于所有 $latex n \geq 1$，只有有限多个 $latex a \in A$ 使得 $latex n + a$ 是无平方的。
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 $latex a < n$.
如果有无限多个 $latex n$ 使得 $latex n + a$ 对于所有 $latex a < n$ 都是无平方的，我们就说 $latex A$ 具有属性 $latex Q$。
How fast must sequences $latex A = \{a_1 < a_2 < \cdots\}$ with properties $latex P$ or $latex Q$ increase?
具有 $latex P$ 或 $latex Q$ 属性的序列 $latex A = \{a_1 < a_2 < \cdots\}$ 必须增加多快？
And so, in October 2025, van Doorn, now back at his day job, left the first comment on the page for Problem 1102 .
因此，2025 年 10 月，回到日常工作的 van Doorn 在页面上针对问题 1102 留下了第一个评论。
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.
Erdős 在 1981 年提出的这个问题询问“无平方”整数集合的性质，即没有重复素因数的整数。
(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.)
（例如，30 是无平方的，因为它等于 2 × 3 × 5，但 18 则不是，因为它等于 2 × 3 × 3；3 重复。）
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.
11 月初，van Doorn 在问题页面上发表了评论，分享了他在不依赖人工智能的情况下找到答案的进展。
Later that day, another commenter on the site replied, claiming he had found a flaw in van Doorn’s argument.
当天晚些时候，该网站的另一位评论者回复称，他发现了范·多恩的论点中的缺陷。
The two traded remarks in rapid succession, and van Doorn convinced his interlocutor that his argument was correct.
两人快速接连地交换意见，范多恩让对话者相信他的论点是正确的。
“I see how your argument works now.
“我现在明白你的论点是如何运作的了。
Nice!” the other mathematician replied.
不错！”另一位数学家回答道。
That other mathematician was Terence 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.
另一位数学家是加州大学洛杉矶分校教授陶哲轩，他可以说是当今在世的最著名的数学家，也无疑是最有影响力的数学家之一。
(Not incidentally, when Tao was just 10 years old , he crossed paths with Erdős.)
（并非偶然，当道只有 10 岁时，他就与埃尔多斯相遇了。）
Bloom’s website, which has the look and feel of an earlier time, was becoming an example of the internet at its democratic best.
布鲁姆的网站具有早期的外观和感觉，正在成为互联网民主最佳状态的一个例子。
“This entire collaboration would not have been possible without Tom’s website and the comments section there,” van Doorn said.
“如果没有汤姆的网站和评论部分，整个合作就不可能实现，”范·多恩说。
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.
无论你是否获得终身教职，无论你是年轻还是年老，无论你是在一所名牌大学还是在大学，都没有关系。
If you wanted to work on math and had good ideas, you could find people to collaborate with.
如果您想从事数学研究并且有好的想法，您可以找到可以合作的人。
But as the winter set in — around the same time that van Doorn found himself collaborating with Terry Tao — things started to change.
但随着冬天的到来——大约在范·多恩发现自己与陶哲轩合作的同时——事情开始发生变化。
Kevin Barreto and Liam Price, both in their early 20s, became friends in the summer of 2025 on a Discord server dedicated to AI.
20 岁出头的 Kevin Barreto 和 Liam Price 于 2025 年夏天在专门用于人工智能的 Discord 服务器上成为了朋友。
Barreto is currently an undergraduate at the University of Cambridge; Price studied some math in college but left before finishing.
巴雷托目前是剑桥大学的本科生；普莱斯在大学学过一些数学，但没读完就离开了。
In December, convinced that the newest AI models might succeed in resolving some Erdős problems, the pair started throwing batches of problems at them.
去年 12 月，两人确信最新的人工智能模型可能会成功解决 Erdő 的一些问题，于是开始向他们抛出一批问题。
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.”
他们很早就意识到，如果他们告诉 GPT-5.2 某个问题的答案未知，那么就不会取得太大进展，因此，正如 Barreto 所说，他们学会了如何“以一种非常特殊的方式提示它，让其认为问题比实际情况更容易”。
Let $latex A \subseteq \mathbb{N}$ be a set of density zero.
设 $latex A \subseteq \mathbb{N}$ 为密度为零的集合。
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 $latex N$?
是否存在 $latex B$ 使得 $latex A \subseteq B + B$ 和 $latex |B \cap \{1, \ldots, N\}| = o(N^{1/2})$ 对于所有大型 $latex N$？
They had what they thought was their first triumph on Erdős Problem 333 .
他们认为这是他们在 Erdős Problem 333 上的第一次胜利。
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.
圣诞节一早，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，这是法学硕士完全自主解决 Erdős 问题的第一个案例，而此前人类从未解决过这个问题。”尽管处理整数集和的 333 并不是一个特别重要的问题，但用人工智能解决它仍然很重要。
But a few hours later, another user pointed out that Erdős himself had provided a resolution to 333 in a paper published in 1977 .
但几个小时后，另一位用户指出，Erdős本人在1977年发表的一篇论文中向333提供了解决方案。
Barreto owned up to the mistake.
巴雷托承认了这个错误。
“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.
他写道：“我对网站所有成员的正式请求是更加注重对当前标记为开放问题的文献检索。”
“As someone who has fallen for this twice now, it’s quite gut-wrenching.”
“作为一个已经两次陷入这种困境的人，这真是令人心痛。”
Let $latex C > 0$ and $latex \epsilon > 0$ be sufficiently small.
令 $latex C > 0$ 和 $latex \epsilon > 0$ 足够小。
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!
是否存在无限多个整数 $latex a, b, n$ 且 $latex a \geq \epsilon n$ 和 $latex b \geq \epsilon n$ 使得 $latex a!b!
\mid n!(a + b – n)!$ and $latex a + b > n + C \log n$?
\mid n!(a + b – n)!$ 和 $latex a + b > n + C \log n$?
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 , a problem about when certain numbers are divisible by other numbers.
他和 Price 没有被吓倒，而是坚持不懈，到 2026 年 1 月 4 日，他们使用 GPT-5.2 Pro 找到了 Erdős 728 的解决方案，这是一个关于某些数字何时能被其他数字整除的问题。
This time nobody could find prior work already proving it.
这次没有人能找到之前的工作已经证明了这一点。
Barreto used another AI tool called Aristotle (developed by a startup called Harmonic) to certify that the proof held together logically.
巴雷托使用了另一种名为亚里士多德（Aristotle）的人工智能工具（由一家名为 Harmonic 的初创公司开发）来证明该证明在逻辑上是一致的。
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 and posted it online.
Nat Sothanaphan 是一名软件工程师，也是唯一比 Bloom、Tao 和 van Doorn 多产的论坛参与者，他让 ChatGPT 写下了正式化的结果并将其发布到网上。
Price developed a methodology for how to ask LLMs to solve open questions.
普莱斯开发了一种如何要求法学硕士解决开放性问题的方法。
First, he would ask a chatbot for a solution.
首先，他会向聊天机器人询问解决方案。
Then he would feed that solution into a fresh instance of the chatbot, asking it to check the previous chatbot’s work.
然后，他将该解决方案输入到聊天机器人的新实例中，要求它检查以前的聊天机器人的工作。
He’d repeat this process until he had what looked like a workable solution.
他会重复这个过程，直到找到看起来可行的解决方案。
(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.)
（这与公司内部一直在做的一些工作相呼应，以创建他们所谓的“工具”或“脚手架”，这些工作可以自动执行 Price 手动进行的迭代。）
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.
Barreto 和 Price 的论文只是过去几个月人工智能至少部分解决的许多 Erdős 问题的一小部分。
There are multiple reasons why these problems in particular have become such a fertile test bed for LLMs.
这些问题之所以成为法学硕士的沃土，有多种原因。
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.
第一个是，总的来说，Erdő 的问题涉及数论、组合学和图论，这些数学领域已被证明比其他领域更容易被大型语言模型所理解。
The problems also vary widely in difficulty and mathematical significance.
这些问题的难度和数学意义也有很大差异。
This variation makes them appropriate for a nascent technology whose abilities also vary widely.
这种变化使它们适合能力也相差很大的新兴技术。
Many of Erdős’ problems had a monetary value attached to them from their moment of inception, a playful incentive from a wandering eccentric.
埃尔多斯的许多问题从一开始就具有货币价值，这是来自流浪怪人的有趣激励。
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.
但现在，随着这些问题已成为人工智能的非正式基准，人们正在根据“每个问题的成本”（解决这些问题所需的代币价格）来讨论他们的解决方案。
Photo by George Csicsery from the documentary N is a Number: A Portrait of Paul Erdős ©1993.
摄影：George Csicsery，摘自纪录片《N 是一个数字：Paul Erdős 的肖像》©1993。
All Rights Reserved.
版权所有。
“A lot of my recent papers should be mostly credited to AI,” van Doorn said.
“我最近的很多论文应该主要归功于人工智能，”范·多恩说。
“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.
“所涉及的想法不是我自己想出来的。”与 Erdős 网站上的许多活跃人士一样，van Doorn 对法学硕士让他能够更快地做更多事情感到兴奋。
“If I read an idea by an LLM, I digest it, try to understand it, simplify it, and generalize it,” he said.
“如果我读到法学硕士的一个想法，我会消化它，尝试理解它，简化它并概括它，”他说。
He uses AI to better understand the math.
他使用人工智能来更好地理解数学。
Not everyone holds themselves to this standard.
并不是每个人都坚持这个标准。
“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.
布鲁姆说：“一个大问题是人工智能被非数学家的人大量使用，他们没有深厚的数学背景，也无法验证输出。”
“They like to move fast, ask their AI to check it, it grows and grows.
“他们喜欢快速行动，让人工智能检查它，它会不断增长。
We’re seeing a lot more of these 100- to 200-page papers that people are posting.
我们看到人们发布了越来越多的 100 到 200 页的论文。
‘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.
‘我解决了这个定理；我让人工智能生成证明、检查证明并撰写论文。”但没有人读过它，也没有人会读它。
It’s a huge challenge now.”
现在这是一个巨大的挑战。”
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 $latex x \to \infty$?
对于任何 $latex x$，如果 $latex A \subset [x, \infty)$ 是一个原始整数集（这样 $latex A$ 中没有不同的元素可以相互分割），那么 $latex \displaystyle\sum_{a \in A} \frac{1}{a \log a} < 1 + o(1),$ 其中 $latex o(1)$ 项 $latex \to 0$ 为 $latex x \to \infty$？
By Price’s own assessment, he doesn’t have enough mathematical understanding to verify the solutions he ultimately coaxes from the LLMs.
根据普莱斯自己的评估，他没有足够的数学理解来验证他最终从法学硕士那里得到的解决方案。
But with Barreto’s help, he’s been able to find mathematicians knowledgeable and willing enough to check the results.
但在巴雷托的帮助下，他能够找到知识渊博且愿意检查结果的数学家。
Both Price and Barreto are co-authors with Tao, Jared Duker Lichtman of Stanford University, and other accomplished mathematicians on a May 2026 paper resolving Erdős Problem 1196 , one of their more significant results.
Price 和 Barreto 与陶、斯坦福大学的 Jared Duker Lichtman 以及其他有成就的数学家在 2026 年 5 月共同发表了一篇论文，解决了 Erdős Problem 1196，这是他们更重要的成果之一。
(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.)
（1196 询问所谓的原始集合的可能大小 - 整数集合，例如 {2, 5, 9, 21}，其中没有数字可以整除任何其他数字。）
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.
布鲁姆感到惊讶的是，尽管 OpenAI、Google DeepMind 和几家初创公司给予了很多关注，但大多数新结果都来自使用公开的法学硕士的业余爱好者和本科生，而不是来自使用更先进的内部模型的企业实验室。
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.
但几周后，即 2026 年 5 月 20 日，情况发生了变化，当时 OpenAI 宣布他们已经解决了最著名的 Erdős 问题之一：单位距离问题。
In the first months of 2026, the major tech companies began to see opportunity in erdosproblems.com.
2026 年的头几个月，各大科技公司开始在 erdosproblems.com 上看到机会。
As Lichtman explained, “Erdős had over 1,000 papers.
正如 Lichtman 解释的那样，“Erdős 拥有 1,000 多篇论文。
They were scattered.” An institute in Hungary had collected scanned images of many of the papers, but nobody had collected all the problems.
他们分散了。”匈牙利的一家研究所收集了许多论文的扫描图像，但没有人收集到所有问题。
“This kind of single repository that anyone can access — labs realized that this could effectively be a benchmark.”
“这种任何人都可以访问的单一存储库——实验室意识到这可以有效地成为一个基准。”
In January, a team of 24 researchers led by Google DeepMind shared a paper 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 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.)
一月份，由 Google DeepMind 领导的一个由 24 名研究人员组成的团队分享了一篇论文，该论文解决了 4 个问题，并找到了另外 9 个问题的旧的、被遗忘的解决方案，此前“使用 Gemini 系统地评估了 Bloom 的 Erdős Problems 数据库中标记为‘开放’的 700 个猜想。”今年 5 月，由 21 名研究人员组成的 DeepMind 团队宣布，“我们最有能力的智能体自主解决了 353 个开放式 Erdős 问题中的 9 个，每个问题的成本为几百美元。” （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以形式逻辑编写的问题。）
And, on May 20, OpenAI shared a solution to the unit distance problem, along with a blog post explaining the work and a companion paper 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).
5 月 20 日，OpenAI 分享了单位距离问题的解决方案，以及一篇解释这项工作的博客文章和一篇配套论文，其中有九位世界级数学家评论了证明的正确性和所做工作的重要性（并提供了结果的简化人类版本）。
Mathematicians had generally believed that Erdős’ conjecture — about how many evenly spaced points can be placed on a plane — was correct.
数学家普遍认为，埃尔多斯的猜想——关于一个平面上可以放置多少个均匀分布的点——是正确的。
To general surprise, OpenAI’s internal model found a counterexample.
令普遍惊讶的是，OpenAI 的内部模型找到了一个反例。
To do so, it had found a sophisticated way to use tools from an area of math called algebraic number theory.
为此，它找到了一种复杂的方法来使用代数数论数学领域的工具。
As Jacob Tsimerman of the University of Toronto wrote in the companion article, “This is a really impressive piece of work.
正如多伦多大学的雅各布·齐默曼（Jacob Tsimerman）在配套文章中所写，“这是一件非常令人印象深刻的作品。
…
……
It is definitely an intimidating construction.”
这绝对是一个令人生畏的建筑。”
In the same article, Tim Gowers 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.
在同一篇文章中，剑桥大学和法兰西学院的蒂姆·高尔斯 (Tim Gowers) 写道，“如果有人写了这篇论文并将其提交给《数学年鉴》，并且有人要求我快速发表意见，我会毫不犹豫地建议接受。
No previous AI-generated proof has come close to that.”
之前没有任何人工智能生成的证据能接近这一点。”
The author on the paper that presented the original solution was given simply as “OpenAI.”
提出原始解决方案的论文作者被简单地称为“OpenAI”。
Terence Tao was 10 years old when he met Erdős.
陶哲轩认识埃尔多斯时才10岁。
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 a version of another long-standing Erdős conjecture.
后来，包括布鲁姆在内的四位数学家小组利用人工智能模型解决单位距离问题的相关技术，反驳了另一个长期存在的埃尔德猜想的一个版本。
The “sum-product” conjecture proposed that if you have sets of numbers, either their sum or their product must grow quickly.
“和积”猜想提出，如果有一组数字，它们的和或它们的乘积必须快速增长。
The mathematicians found a set of real numbers for which both the sum and the product grow more slowly than expected.
数学家发现了一组实数，其总和和乘积的增长速度都比预期慢。
The conjecture for integers remains open.
整数猜想仍然悬而未决。
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.
弄清楚人工智能将对数学和数学家产生什么影响意味着不仅要关注其最重要的结果，还要研究它如何改变解决日常问题的日常实践。
Noga Alon, the Princeton mathematician, estimates that he has solved a few dozen Erdős problems over his career.
普林斯顿大学数学家诺加·阿隆估计，在他的职业生涯中，他已经解决了几十个埃尔多斯问题。
He has now stopped trying.
他现在已经停止尝试了。
“Once AI started to solve them, there is no point anymore,” he said.
“一旦人工智能开始解决这些问题，就没有任何意义了，”他说。
Terry Tao has stepped away from the Erdős problem community to focus on getting work done.
陶哲轩 (Terry Tao) 已离开 Erdős 问题社区，专注于完成工作。
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.
范·杜恩目前仍在一家客户服务公司从事日常工作，他表示法学硕士“显然比我更善于思考和做数学。
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.
我无法与当前的人工智能系统相提并论。”然而，他补充道，“我最终写出的证明比 ChatGPT 提出的证明更简单、更通用，也更容易被其他人阅读。”人工智能无疑提高了他的工作效率，而且他仍然乐在其中。
“If you want to play piano, you aren’t going to hire a piano-playing machine that does it better than you.
“如果你想弹钢琴，你就不会雇佣一台比你弹得更好的钢琴弹奏机。
You will play the piano because you like playing the piano.
你会弹钢琴，因为你喜欢弹钢琴。
I enjoy thinking about numbers, doing math, writing papers.
我喜欢思考数字、做数学、写论文。
I’m not going to hire a paper-making machine that does it for me.”
我不会雇佣一台造纸机来帮我做这件事。”
For van Doorn, there is joy to be found in digesting the responses he gets from LLMs.
对于 van Doorn 来说，消化法学硕士的回复是一种乐趣。
“I’ve been doing a lot of math recently thanks to the Erdős-problems community.
“感谢 Erdős 问题社区，我最近做了很多数学工作。
It used to be the case I just did everything all by myself, struggled alone in my room.
以前都是我一个人做所有的事，一个人在房间里挣扎。
I don’t know how it happened, but nowadays people contact me saying, ‘I have this idea.
我不知道这是怎么发生的，但现在人们联系我说，‘我有这个想法。
Do you want to join me thinking about this?’”
你想和我一起思考这个问题吗？”
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.
人工智能能力的增强使得像普莱斯或范·多恩这样的人在数学训练较少的情况下更容易解决谜题，同时也让像阿隆这样一生致力于理解这些谜题的人变得不那么感兴趣。
Nonetheless, “many and maybe most good mathematicians will use AI,” Alon said.
尽管如此，“许多，也许是大多数优秀的数学家都会使用人工智能，”阿隆说。
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 , the highest honor in math, he announced that he was leaving academia for a job at OpenAI.
他指出，许多一流的数学家已经离开学术界去人工智能公司工作，不仅因为他们这样做的报酬很高，而且因为“也许这就是现在的行动”。 2026 年 7 月，齐默尔曼获得数学界最高荣誉菲尔兹奖的同一天，他宣布离开学术界前往 OpenAI 工作。
What Is the Unit Distance Problem?
什么是单位距离问题？
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.
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]:相对 [&_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;"> 假设您想要在平面上排列点以创建点对它们之间的距离相同。
How can you create the most pairs?
如何创造最多的配对？
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.
如果你排列你的点，使它们形成正方形和五边形等多边形，你将得到与你拥有的点相同数量的对。
But if you put points in a lattice, you can create substantially more equidistant pairs.
但如果将点放入格子中，则可以创建更多等距对。
Erdős conjectured that it wasn’t possible to do substantially better than this.
埃尔多斯推测，不可能做得比这更好。
For 80 years, mathematicians generally believed he was correct.
80 年来，数学家们普遍相信他是正确的。
But Erdős was mistaken.
但埃尔多斯错了。
OpenAI found a pattern (similar to the one shown below) that, for a given number of points, produces more pairs than a lattice can.
OpenAI 发现了一种模式（类似于下图所示），对于给定数量的点，它可以产生比晶格更多的对。
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《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。
Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.
辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。
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版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。