近八十年来，数学家们一直在研究一个看似简单的问题：如果在平面上放置 \(n\) 个点，那么有多少对点之间的距离恰好为 \(1\)？
这就是平面单位距离问题，最早由保罗·埃尔德什于1946年提出。它是组合几何学中最著名的问题之一，表述简单，却极难解决。2005年由布拉什、莫泽和帕赫合著的《离散几何研究问题》一书称其为“可能是组合几何学中最著名（也最容易解释）的问题”。普林斯顿大学著名组合学家诺加·阿隆将其描述为“埃尔德什最喜爱的问题之一”。埃尔德什甚至为攻克该问题设立了现金悬赏。
今天，我们分享一项关于单位距离问题的突破性进展。自埃尔德什的原创工作以来，学界普遍认为下文将进一步展示的“方格”构造在最大化单位距离对数方面基本是最优的。一个OpenAI内部模型推翻了这一长期存在的猜想，提供了一组无限多的新范例，实现了多项式级别的改进。该证明已由一组外部数学家审核。他们还撰写了一篇配套论文，详细阐述论证过程，并为该结果的重要性提供了进一步的背景和语境。
这一结果的发现方式同样引人注目。该证明来自一个全新的通用推理模型，而非一个专门为数学训练、为搜索证明策略而搭建、或专门针对单位距离问题的系统。作为测试先进模型能否为前沿研究做出贡献的更广泛努力的一部分，我们在一系列埃尔德什问题上对其进行了评估。在此案例中，它产出了一个证明，解决了一个悬而未决的问题。
这一证明是数学界和 AI 社区的一个重要里程碑。它标志着，一个在数学子领域中占据核心地位的著名开放问题，首次由 AI 自主解决。同时，它也展示了当前这些系统所支持的推理深度。数学为推理提供了一个尤为清晰的试验场：问题精确，潜在证明可被验证，而一个长篇论证只有在推理从头到尾都站得住脚时才算成立。解决该问题的方法同样值得关注。该证明将代数数论中意想不到的深刻思想，巧妙地应用于一个基础的几何问题。
菲尔兹奖得主蒂莫西·高尔斯在随附论文中称这一成果是“AI 数学领域的一个里程碑”。著名数论学家阿鲁尔·尚卡尔表示：“在我看来，这篇论文表明，当前的 AI 模型已不仅仅是人类数学家的助手——它们能够产生独创性的巧妙想法，并将其付诸实践直至成功。”
数学家们对该成果的评价
“这是埃尔德什最喜爱的问题之一，我曾亲耳听他在讲座中多次提及。我认为可以公平地说，每一位从事组合几何研究的数学家都思考过这个问题，而许多其他领域的数学家也至少花过一些时间琢磨它……在我看来，OpenAI 的内部模型解决这个问题是一项卓越的成就，它终结了一个长期悬而未决的开放问题。正确答案并非 $n^{1 + o \left(\right. 1 \left.\right)}$n 1+o(1) 这一点令人惊讶，而该构造及其分析过程，则以一种优雅而巧妙的方式运用了代数数论中相当精深的工具。”
诺加·阿隆
“毫无疑问，单位距离问题的解决是 AI 数学领域的一个里程碑：如果这篇论文是由人类撰写并提交给《数学年刊》，而有人请我给出快速意见，我会毫不犹豫地建议接收。此前没有任何 AI 生成的证明能达到这个水平。”
蒂莫西·高尔斯
“该模型的思维链非常引人入胜。值得注意的是，其绝大多数思考过程都试图为广为人知的上界构造反例，而非试图证明该上界。这表明该模型兼具良好的直觉、敢于尝试学界认为希望渺茫的方法的意愿，以及倾向于进行构造性尝试的特质……在我看来，这篇论文证明了当前 AI 模型已超越人类数学家的助手角色——它们能够产生独创性的巧妙想法，并将其付诸实践并取得成果。”
Arul Shankar
“这是一项令人印象极为深刻的工作，我会毫不犹豫地接受它发表在任何期刊上。实际上，我曾短暂研究过这个问题，并试图构造一个反例，但未能取得进展……即便你清楚其中的原理，要彻底理解这个构造过程也相当令人望而生畏，而亲自去探索则更加困难。”
Jacob Tsimerman
Noga Alon Tim Gowers Arul Shankar Jacob Tsimerman
Noga Alon Tim Gowers Arul Shankar Jacob Tsimerman
该证明可在此处获取。由顶尖外部数学家撰写的配套论文可在此处获取。该模型思维链的删节版可在此处获取。
此前已知的、通过缩放正方形网格构造多个单位距离的方法。
单位距离问题
设 \( u(n) \) 为平面上 \( n \) 个点之间可能达到的最大单位距离对数。构造达到线性增长率的例子很容易：将 \( n \) 个点排成一条直线可得到 \( n-1 \) 对，而正方形网格则可得到约 \( 2n \) 对。此前已知的最佳构造来自缩放后的正方形网格，结果甚至更好：对于某个常数 \( C \)，可达到 \( n^{1 + C / \log \log (n)} \) 对。由于 \( \log \log (n) \) 随 \( n \) 趋于无穷，指数中的附加项趋于 \( 0 \)，这意味着这些构造的增长速度仅略快于线性。几十年来，人们普遍认为这个速率基本上是最优的，没有任何构造能显著超越正方形网格。用专业术语来说，Erdős 猜想的上界为 \( n^{1 + o(1)} \)，其中附加的 \( o(1) \) 表示一个随 \( n \) 趋于 \( 0 \) 的项。
我们的新结果推翻了这一猜想。更准确地说，对于无穷多个 \( n \) 值，该证明构造了 \( n \) 个点的配置，其中至少包含 \( n^{1 + \delta} \) 个单位距离对，且 \( \delta > 0 \) 为某个固定指数。（最初的 AI 证明并未给出明确的 \( \delta \) 值，但普林斯顿大学数学教授 Will Sawin 即将发表的一项改进表明，可以取 \( \delta = 0.014 \)。）
该问题的历史有助于理解为何这一结果令人惊讶。自 1946 年 Erdős 最初构造以来，已知的最佳下界基本未变。最佳上界 $O(n^{4/3})$ 可追溯到 1984 年 Spencer、Szemerédi 和 Trotter 的工作，尽管后来 Székely、Katz 与 Silier、Pach、Raz、Solymosi 等人进行了改进及相关结构性研究，该上界仍基本保持不变。作为支持该猜想的证据，Matoušek 以及 Alon-Bucić-Sauermann 研究了平面中非欧几里得距离的问题，并证明这些非欧几里得距离在某种意义上“大多数”都符合该猜想。
令人惊讶的是，该构造的关键要素来自数学中一个截然不同的分支——代数数论，它研究整数扩展（即代数数域）中的因式分解等概念。
在验证了初始证明后，我们研究了模型在该问题上随测试时计算量变化的成功率。结果如下所示。
来自代数数论的新技术
从宏观层面看，该证明始于一个熟悉的几何思路，并将其推向了一个意想不到的方向。
Erdős 的原始下界可以通过高斯整数来理解：即形如 $a + bi$ 的数，其中 $a$ 和 $b$ 是整数，$i$ 是 $-1$ 的平方根。高斯整数扩展了普通整数，并且与普通整数一样，具有唯一质因数分解等性质。这种对普通整数或有理数的扩展被称为代数数域。新的论证用代数数论中更复杂的推广替代了高斯整数，这些推广具有更丰富的对称性，能够产生更多单位长度的差值。
该精确论证使用了无限类域塔和 Golod–Shafarevich 理论等工具，以证明论证所需的数域确实存在。这些概念对代数数论学者而言是众所周知的，但令人大为意外的是，这些概念竟对欧几里得平面中的几何问题具有启示意义。
这对数学意味着什么
这一结果标志着人工智能与数学交互中的一个重要时刻：一个AI系统自主解决了一个活跃领域中心长期悬而未决的开放性问题。它也让我们得以一窥AI与人类数学家之间一种新型合作方式的雏形。在此案例中，外部数学家撰写的配套论文所描绘的图景，远比原始解决方案本身要丰富得多。
正如托马斯·布鲁姆在配套论文中所写：
“在评估一个AI生成的证明的重要性和影响力时，我问自己的一个问题是：它是否让我们对这个问题有了新的认识？我们现在是否更好地理解了离散几何？我认为答案是有所肯定的：这表明，数论构造方法在这类问题上的解释力远超我们之前的预期；此外，所需的数论知识可能非常深奥。毫无疑问，未来几个月，许多代数数论学家将密切关注离散几何中的其他开放问题。”
该解决方案揭示的代数数论与离散几何之间意想不到的联系，正是这一结果引人注目的部分原因。它不仅解决了一个具体的猜想，还可能为数学家们提供一座桥梁，去探索更多相关的衍生问题。
布鲁姆还指出了更广泛的可能性：
“知识的边界是参差不齐的。毫无疑问，未来数月乃至数年，数学的许多其他领域也将取得类似的成功——AI通过揭示意想不到的联系，并将现有技术手段推向极限，来解决长期悬而未决的开放性问题。AI正在帮助我们更充分地探索人类数个世纪以来构建的数学大教堂；还有哪些未被发现的奇观正在幕后等待？”
这一结果提供了一个充满希望的范例：AI不仅贡献了一个解决方案，更带来了一项数学发现，其重要性通过后续的人类理解而变得更加清晰和丰富。
为何这很重要
这一结论的意义远不止于该特定成果。更强的数学推理能力能让 AI 成为更出色的研究伙伴：它能够串联起复杂的思维线索，将不同知识领域的想法联系起来，揭示专家可能尚未重视的、有前景的研究路径，并帮助研究人员在那些原本因过于复杂或耗时而难以攻克的问题上取得进展。
这些能力的重要性并不局限于数学领域。如果一个模型能够保持复杂论证的一致性，将不同知识领域的想法联系起来，并产出经得起专家审视的成果，那么这些能力在生物学、物理学、材料科学、工程学和医学领域同样有用，并且它们也是我们迈向更自动化研究的长期路径的一部分：即能够帮助科学家和工程师探索更多想法、攻克更难技术问题的系统。
AI 即将在研究中的创造性部分扮演非常重要的角色，而最重要的则是 AI 研究本身。尽管这一进展并非出人意料，但它强化了我们的紧迫感——我们需要理解 AI 发展的这一新阶段、对齐高度智能系统所面临的挑战，以及人机协作的未来。
那个未来仍然取决于人类的判断。专业知识的价值只会变得更高，而非更低。AI 可以帮助搜索、建议和验证。而由人类来选择那些真正重要的问题、解读结果，并决定下一步要探索哪些问题。
For nearly 80 years, mathematicians have studied a deceptively simple question: if you place $n$n points in the plane, how many pairs of points can be exactly distance $1$1 apart?
This is the planar unit distance problem, first posed by Paul Erdős in 1946. It is one of the best-known questions in combinatorial geometry, easy to state and remarkably difficult to resolve. The 2005 book Research Problems in Discrete Geometry, by Brass, Moser, and Pach, calls it “possibly the best known (and simplest to explain) problem in combinatorial geometry.” Noga Alon, a leading combinatorialist at Princeton, describes it as “one of Erdős’ favorite problems.” Erdős even offered a monetary prize for resolving this problem.
Today, we share a breakthrough on the unit distance problem. Since Erdős’s original work, the prevailing belief has been that the “square grid” constructions depicted further below were essentially optimal for maximizing the number of unit-distance pairs. An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement. The proof has been checked by a group of external mathematicians. They have also written a companion paper explaining the argument and providing further background and context for the significance of the result.
The result is also notable for how it was found. The proof came from a new general-purpose reasoning model, rather than from a system trained specifically for mathematics, scaffolded to search through proof strategies, or targeted at the unit distance problem in particular. As part of a broader effort to test whether advanced models can contribute to frontier research, we evaluated it on a collection of Erdős problems. In this case, it produced a proof resolving the open problem.
This proof is an important milestone for the math and AI communities. It marks the first time that a prominent open problem, central to a subfield of mathematics, has been solved autonomously by AI. It also demonstrates the depth of reasoning these systems now support. Mathematics provides a particularly clear testbed for reasoning: the problems are precise, potential proofs can be checked, and a long argument only works if the reasoning holds together from beginning to end. The method by which the problem was solved is also notable. The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question.
Fields medalist Tim Gowers, writing in the companion paper, calls the result “a milestone in AI mathematics.” According to leading number theorist Arul Shankar, “In my opinion this paper demonstrates that current AI models go beyond just helpers to human mathematicians – they are capable of having original ingenious ideas, and then carrying them out to fruition”.
Mathematicians on the result
“This has been one of Erdős' favorite problems, I have heard him myself mentioning the problem multiple times in his lectures. I believe it would be fair to say that every mathematician working in Combinatorial Geometry thought about this problem, and lots of mathematicians working in other areas spent at least some time thinking about it… The solution of the problem by the internal model of Open AI is, in my opinion, an outstanding achievement, settling a long-standing open problem. The fact that the correct answer is not $n^{1 + o \left(\right. 1 \left.\right)}$n 1+o(1)is surprising, and the construction and its analysis apply fairly sophisticated tools from algebraic number theory in an elegant and clever way.”
Noga Alon
“There is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics: 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.”
Tim Gowers
“The model’s CoT is deeply interesting. It is noteworthy that a significant majority of the thoughts are trying to construct a counterexample to the widely believed upper bound, rather than trying to prove it. This argues that the model has some combination of good intuition, willingness to try approaches considered long-shot by the community, and a predisposition to attempt constructions.… In my opinion this paper demonstrates that current AI models go beyond just helpers to human mathematicians – they are capable of having original ingenious ideas, and then carrying them out to fruition.”
Arul Shankar
“This is a really impressive piece of work, and I would accept it for any journal without hesitation. I actually briefly worked on this problem and tried to make a counterexample, but failed to make progress… It is definitely an intimidating construction to see through even if you know what is going on, and even harder to go play for yourself.”
Jacob Tsimerman
Noga Alon Tim Gowers Arul Shankar Jacob Tsimerman
Noga Alon Tim Gowers Arul Shankar Jacob Tsimerman
The proof is available here . The companion paper by leading external mathematicians is available here . You can find an abridged version of the model’s chain of thought here .
Previously known construction of many unit distances from a rescaled square grid.
The unit distance problem
Let $u \left(\right. n \left.\right)$u(n) be the largest possible number of unit-distance pairs among $n$n points in the plane. Examples attaining linear growth rate are easy to construct: placing $n$n points in a line gives $n - 1$n−1 pairs, while a square grid gives about $2 n$2 n pairs. The previously best known construction, coming from a rescaled square grid, turns out to give even more: $n^{1 + C / log log \left(\right. n \left.\right)}$n 1+C/l o g l o g(n) for a constant $C$C. Since $log log \left(\right. n \left.\right)$lo g lo g(n) tends to infinity with $n$n, the additional term in the exponent tends to $0$0, meaning these constructions achieve growth only slightly faster than linear. For decades, it was widely believed that this rate was essentially the best possible, and no construction could improve significantly over the square grid. In technical terms, Erdős conjectured an upper bound of $n^{1 + o \left(\right. 1 \left.\right)}$n 1+o(1) in which the additional $o \left(\right. 1 \left.\right)$o(1) indicates a term tending to $0$0 with $n$n.
Our new result disproves this conjecture. More precisely, for infinitely many values of $n$n, the proof constructs configurations of $n$n points with at least $n^{1 + \delta}$n 1+δ unit-distance pairs, for some fixed exponent $\delta > 0$δ>0. (The original AI proof does not give an explicit $\delta$δ, but a forthcoming refinement due to Princeton mathematics professor Will Sawin has shown one can take $\delta = 0.014$δ=0.014.)
The history of the problem helps to see why the result is surprising. The best known lower bound had been essentially unchanged since Erdős’s original 1946 construction. The best upper bound, $O \left(\right. n^{4 / 3} \left.\right)$O(n 4/3), dates to work by Spencer, Szemerédi, and Trotter in 1984, and despite later refinements and related structural work by Székely, Katz and Silier, Pach, Raz, and Solymosi and by others, the upper bound has remained essentially unchanged. As evidence in favor of the conjecture, Matoušek and Alon-Bucić-Sauermann studied the problem with non-Euclidean distances in the plane, and proved that "most" of these non-Euclidean distances obey the conjecture in some sense.
Surprisingly, the key ingredients of the construction come from a very different part of mathematics known as algebraic number theory, which studies concepts like factorization in extensions of the integers known as algebraic number fields.
After verifying the initial proof, we investigated the success rate of our models on this problem with varying amounts of test-time compute. The results are shown here.
New techniques from algebraic number theory
At a high level, the proof begins with a familiar geometric idea and pushes it in an unexpected direction.
Erdős’s original lower bound can be understood through the Gaussian integers: numbers of the form $a + b i$a+bi, where $a$a and $b$b are integers and $i$i is the square root of $- 1$−1. The Gaussian integers extend the ordinary integers and, like them, enjoy properties such a unique factorization into primes. Such extensions of the ordinary integers or rationals are known as algebraic number fields. The new argument replaces the Gaussian integers by more complicated generalizations from algebraic number theory with richer symmetries that can create many more unit-length differences.
The precise argument uses tools such as infinite class field towers and Golod–Shafarevich theory to show the number fields required for the argument actually exist. These ideas were well-known to algebraic number theorists, but it came as a great surprise that these concepts have implications for geometric questions in the Euclidean plane.
What this means for mathematics
This result marks an important moment in the interaction between AI and mathematics: an AI system has autonomously resolved a longstanding open problem at the center of an active field. It also offers an early glimpse of a new kind of collaboration between AI and human mathematicians. In this case, the companion work by external mathematicians paints a substantially richer picture than the original solution alone.
As Thomas Bloom writes in the companion note:
“When assessing the importance and influence of an AI-generated proof, a question I ask myself is: has this taught us something new about the problem? Do we understand discrete geometry better now? I think the answer is a moderated yes: this shows that there is a lot more that number theoretic constructions have to say about these sorts of questions than we suspected; moreover, that the number theory required can be very deep. No doubt many algebraic number theorists will be taking a close look at other open problems in discrete geometry in the coming months.”
The unexpected connection between algebraic number theory and discrete geometry revealed by the solution is part of what makes the result notable. It does not simply settle a specific conjecture, but may provide mathematicians with a bridge to begin exploring further related problems.
Bloom also points toward a broader possibility:
“The frontiers of knowledge are very spiky, and no doubt the coming months and years will see similar successes in many other areas of mathematics, where long-standing open problems are resolved by an AI revealing unexpected connections and pushing the existing technical machinery to its limit. AI is helping us to more fully explore the cathedral of mathematics we have build over the centuries; what other unseen wonders are waiting in the wings?”
This result provides a promising example: AI contributing not only a solution, but a mathematical discovery whose significance becomes clearer and richer through subsequent human understanding.
Why this matters
The takeaway is bigger than this particular result. Better mathematical reasoning can make AI a stronger research partner: something that can hold together difficult lines of thought, connect ideas across distant areas of knowledge, surface promising paths experts may not have prioritized, and help researchers make progress on problems that would otherwise be too complex or time-intensive to tackle.
Those capabilities matter beyond mathematics. If a model can keep a complicated argument coherent, connect ideas across distant areas of knowledge, and produce work that survives expert scrutiny, those are also useful abilities in biology, physics, materials science, engineering, and medicine, and they are part of our longer-term path toward more automated research: systems that can help scientists and engineers explore more ideas and pursue harder technical questions.
AI is about to start taking a very serious role in the creative parts of research, and most importantly AI research itself. While this progress is not unexpected, it reinforces the urgency we feel about understanding this next phase of AI development, the challenges of aligning very intelligent systems, and the future of human-AI collaboration.
That future still depends on human judgment. Expertise becomes more valuable, not less. AI can help search, suggest, and verify. People choose the problems that matter, interpret the results, and decide what questions to pursue next.