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The planar unit distance problem explores the maximum number of line segments of equal length that can connect points on an infinite plane.
On May 20, OpenAI announced that its AI chatbot had disproven a claim put forward by Hungarian mathematician Paul Erdős (1913–1996) regarding the so-called unit distance problem. Drawing on hints from mathematicians, the AI cracked this geometry puzzle that has stood unsolved for 80 years. The finding has since been independently verified by mathematicians unaffiliated with OpenAI.
In 1946, Erdős proposed what he believed to be the optimal arrangement of points on a plane to maximize pairs of points separated by a fixed distance. He further issued a challenge, asserting that no one could devise a better solution.
Now OpenAI states its system has accomplished exactly that, leveraging techniques from algebraic number theory to select points whose coordinates are solutions to specific equations. The breakthrough has taken the mathematics community by surprise.
"If Erdős were alive today, he would absolutely be overjoyed by this progress," said Tom Trotter, a mathematician at Georgia Tech who co-authored papers with Erdős on related topics.
Sebastien Bubeck, a mathematician at OpenAI, remarked that this marks the first major scientific result independently produced by an AI in a research field. Tony Feng, a mathematician at the University of California, Berkeley, commented: "I have long been cautious about AI’s impact on mathematics, but this achievement is truly incredible."
Daniel Litt, a mathematician at the University of Toronto, was among the independent researchers invited by OpenAI to validate the proof. "This is the first fully AI-generated research finding that stands on its own as a substantial contribution to academia," he noted.
In geometry, points can be arranged on a plane to create numerous pairs with identical spacing. For instance, a regular nonagon (nine-sided polygon) features nine pairs of equidistant points due to its equal side lengths. By contrast, placing nine points on a square grid yields twelve such pairs.
Erdős demonstrated that expanding grids can contain vast numbers of equidistant point pairs, with the number of such pairs growing slightly faster than the total number of points as the grid extends infinitely. He also conjectured that no arrangement could outperform his approach in maximizing equidistant point pairs.
OpenAI has now proven otherwise. Its AI model adopted methodologies from algebraic number theory to identify points with coordinates derived from solutions to targeted equations. Bubeck explained that the model produced an extensive chain of reasoning. It was given an open-ended prompt asking whether Erdős’ conjecture held true, rather than a direct instruction to disprove it. Mehtaab Sawhney, another mathematician at OpenAI, said: "It is astonishing to watch the model reason through a problem just like a human being."
The full reasoning is documented in a 125-page report, which OpenAI has not released in its entirety. The company has also declined to disclose the model’s official name. Bubeck described it as an experimental general reasoning model, not custom-built for mathematics. It completed the entire work autonomously based on a prompt that simply restated Erdős’ problem.
This approach differs fundamentally from the conventional "orchestrated" method of using AI for mathematical problem-solving. The latter requires researchers to guide large language models (LLMs) through repeated iterations to correct errors and work toward a solution.
In comparison, OpenAI’s system delivers consistent results regardless of how the prompt is phrased. Many top AI solutions for mathematical problems to date rely heavily on trial and error, where crafting effective prompts has become a specialized skill. "Now you can phrase the question in virtually any way, and the model will understand it correctly," Bubeck said.
Litt pointed out that the algorithm powering the AI’s solution stems from algebraic number theory, showing that AI models are breaking out of narrow specialized silos to achieve broader applicability. He added that no human could master mathematical literature as comprehensively as an LLM.
"We all expected to see something like this eventually, but not this soon," said Mark Sellke, a mathematician at OpenAI. "This represents a massive leap forward from what we were accustomed to just a month ago."
By Li Muzi
Source: China Science News (May 25, 2026, International Edition, Page 2)
AI破解80年前数学难题
平面单位距离问题探讨的是，在无限大的纸上，最多可以画出多少条等长的线段来连接纸上的点。
5月20日，OpenAI宣布，它的人工智能（AI）聊天机器人在所谓的单位距离问题上，证明了匈牙利数学家Paul Erdos（1913—1996年）的观点是错误的。OpenAI的AI聊天机器人利用来自数学家的一个提示，破解了这道已有80年历史的几何难题。目前，这一发现已得到与该公司无关的数学家的独立验证。
1946年，Erdos推导出了一个他认为的平面上的点的最佳排列方式，即让尽可能多的点之间的距离保持在给定距离上。他还提出了一个挑战：没有人能够做得更好。
现在，OpenAI表示，他们的系统已经做到了这一点。它是通过运用代数数论中的相关技术实现的。这使得它能够选取坐标值作为特定方程解的点。这一发现令数学家感到震惊。
“如果Erdos还活着，他肯定会对这一进展欣喜若狂。”美国佐治亚理工学院的数学家Tom Trotter说。他曾与Erdos共同撰写过相关论文。
OpenAI的数学家Sebastien Bubeck表示，他认为这是AI首次在一个科研领域自主产生的重要成果。美国加利福尼亚大学伯克利分校的数学家Tony Feng表示：“我一向对AI在数学领域的影响持审慎态度，但这次的成果实在令人难以置信。”
加拿大多伦多大学的数学家Daniel Litt是OpenAI邀请来验证这一证明的独立研究人员之一。他表示：“这是第一个完全由AI独立得出的研究结果，其本身就极具研究价值。”
在几何学中，点可以在平面上进行排列，并让许多对点具有相同的相互距离。例如，一个有9条边的正九边形就有9组等距的点对，这是因为9条边的边长完全相等；而在一个正方形网格上放置9个点，则能形成12组这样的等距点对。
Erdos证明了越来越大的网格如何能够包含大量距离相等的点，并且这个网格会以比点数增长略快的速度无限延伸。此外，他还提出一个猜想，即没有人能找到一种更好的方法来排列这么多的具有相同距离的点。
然而OpenAI表示已经具备了这种能力。该公司的AI模型利用代数数论中的技术实现了这一目标，该技术使它能够选择坐标作为特定方程解的点。Bubeck说，该模型已生成了一条很长的思维链，使得得出这一答案的提示是一个关于Erdos的猜想是真还是假的开放式问题，并不是一个证明他是错误的明确要求。OpenAI的数学家Mehtaab Swahney说：“看到这个模型像人类一样真正通过推理来解决问题，着实令人惊叹。”
这一推理过程包含在一份长达125页的文件中，但OpenAI尚未完全公布这份文件。此外，该公司也未透露其模型的具体名称。Bubeck表示，这是一个实验性的通用推理模型，并非专为解决数学问题而设计，并且它能够根据一个提示自主完成所有工作，即对Erdos问题给出一个机器重写的表述。
Bubeck表示，这种做法与利用AI解决数学问题的“编排”方法截然不同。在“编排”方法中，研究人员会让大语言模型（LLM）通过不断迭代的方式纠正自身的错误，从而找到问题的解决方案。
相比之下，OpenAI系统给出的答案不会因提示语的表述方式不同而有太大差异。到目前为止，一些针对数学问题的最佳AI解决方案都需要大量试错，而提示语的使用已成为一门艺术。“如今，你基本可以任何你想要的方式提出问题，而模型都会正确理解这些问题。”Bubeck说。
Litt表示，由AI生成的解决方案所采用的算法来自代数数论这一事实表明，AI模型正在超越专业化“孤岛”的局限，实现更广泛的应用。他补充道，没有人能够像LLM那样全面掌握数学领域的文献内容。
“我们所有人都曾预料有朝一日会看到这样的情况，但没想到会这么快。”OpenAI的数学家Mark Sellke说，“这与一个月前我们习以为常的情况相比，是一个巨大的飞跃。”（李木子）
来源：《中国科学报》 (2026-05-25 第2版 国际)
AI Solves an 80-Year-Old Mathematical Puzzle
The planar unit distance problem explores the maximum number of line segments of equal length that can connect points on an infinite plane.
On May 20, OpenAI announced that its AI chatbot had disproven a claim put forward by Hungarian mathematician Paul Erdős (1913–1996) regarding the so-called unit distance problem. Drawing on hints from mathematicians, the AI cracked this geometry puzzle that has stood unsolved for 80 years. The finding has since been independently verified by mathematicians unaffiliated with OpenAI.
In 1946, Erdős proposed what he believed to be the optimal arrangement of points on a plane to maximize pairs of points separated by a fixed distance. He further issued a challenge, asserting that no one could devise a better solution.
Now OpenAI states its system has accomplished exactly that, leveraging techniques from algebraic number theory to select points whose coordinates are solutions to specific equations. The breakthrough has taken the mathematics community by surprise.
"If Erdős were alive today, he would absolutely be overjoyed by this progress," said Tom Trotter, a mathematician at Georgia Tech who co-authored papers with Erdős on related topics.
Sebastien Bubeck, a mathematician at OpenAI, remarked that this marks the first major scientific result independently produced by an AI in a research field. Tony Feng, a mathematician at the University of California, Berkeley, commented: "I have long been cautious about AI’s impact on mathematics, but this achievement is truly incredible."
Daniel Litt, a mathematician at the University of Toronto, was among the independent researchers invited by OpenAI to validate the proof. "This is the first fully AI-generated research finding that stands on its own as a substantial contribution to academia," he noted.
In geometry, points can be arranged on a plane to create numerous pairs with identical spacing. For instance, a regular nonagon (nine-sided polygon) features nine pairs of equidistant points due to its equal side lengths. By contrast, placing nine points on a square grid yields twelve such pairs.
Erdős demonstrated that expanding grids can contain vast numbers of equidistant point pairs, with the number of such pairs growing slightly faster than the total number of points as the grid extends infinitely. He also conjectured that no arrangement could outperform his approach in maximizing equidistant point pairs.
OpenAI has now proven otherwise. Its AI model adopted methodologies from algebraic number theory to identify points with coordinates derived from solutions to targeted equations. Bubeck explained that the model produced an extensive chain of reasoning. It was given an open-ended prompt asking whether Erdős’ conjecture held true, rather than a direct instruction to disprove it. Mehtaab Sawhney, another mathematician at OpenAI, said: "It is astonishing to watch the model reason through a problem just like a human being."
The full reasoning is documented in a 125-page report, which OpenAI has not released in its entirety. The company has also declined to disclose the model’s official name. Bubeck described it as an experimental general reasoning model, not custom-built for mathematics. It completed the entire work autonomously based on a prompt that simply restated Erdős’ problem.
This approach differs fundamentally from the conventional "orchestrated" method of using AI for mathematical problem-solving. The latter requires researchers to guide large language models (LLMs) through repeated iterations to correct errors and work toward a solution.
In comparison, OpenAI’s system delivers consistent results regardless of how the prompt is phrased. Many top AI solutions for mathematical problems to date rely heavily on trial and error, where crafting effective prompts has become a specialized skill. "Now you can phrase the question in virtually any way, and the model will understand it correctly," Bubeck said.
Litt pointed out that the algorithm powering the AI’s solution stems from algebraic number theory, showing that AI models are breaking out of narrow specialized silos to achieve broader applicability. He added that no human could master mathematical literature as comprehensively as an LLM.
"We all expected to see something like this eventually, but not this soon," said Mark Sellke, a mathematician at OpenAI. "This represents a massive leap forward from what we were accustomed to just a month ago."
By Li Muzi
Source: China Science News (May 25, 2026, International Edition, Page 2)
AI破解80年前数学难题
平面单位距离问题探讨的是，在无限大的纸上，最多可以画出多少条等长的线段来连接纸上的点。
5月20日，OpenAI宣布，它的人工智能（AI）聊天机器人在所谓的单位距离问题上，证明了匈牙利数学家Paul Erdos（1913—1996年）的观点是错误的。OpenAI的AI聊天机器人利用来自数学家的一个提示，破解了这道已有80年历史的几何难题。目前，这一发现已得到与该公司无关的数学家的独立验证。
1946年，Erdos推导出了一个他认为的平面上的点的最佳排列方式，即让尽可能多的点之间的距离保持在给定距离上。他还提出了一个挑战：没有人能够做得更好。
现在，OpenAI表示，他们的系统已经做到了这一点。它是通过运用代数数论中的相关技术实现的。这使得它能够选取坐标值作为特定方程解的点。这一发现令数学家感到震惊。
“如果Erdos还活着，他肯定会对这一进展欣喜若狂。”美国佐治亚理工学院的数学家Tom Trotter说。他曾与Erdos共同撰写过相关论文。
OpenAI的数学家Sebastien Bubeck表示，他认为这是AI首次在一个科研领域自主产生的重要成果。美国加利福尼亚大学伯克利分校的数学家Tony Feng表示：“我一向对AI在数学领域的影响持审慎态度，但这次的成果实在令人难以置信。”
加拿大多伦多大学的数学家Daniel Litt是OpenAI邀请来验证这一证明的独立研究人员之一。他表示：“这是第一个完全由AI独立得出的研究结果，其本身就极具研究价值。”
在几何学中，点可以在平面上进行排列，并让许多对点具有相同的相互距离。例如，一个有9条边的正九边形就有9组等距的点对，这是因为9条边的边长完全相等；而在一个正方形网格上放置9个点，则能形成12组这样的等距点对。
Erdos证明了越来越大的网格如何能够包含大量距离相等的点，并且这个网格会以比点数增长略快的速度无限延伸。此外，他还提出一个猜想，即没有人能找到一种更好的方法来排列这么多的具有相同距离的点。
然而OpenAI表示已经具备了这种能力。该公司的AI模型利用代数数论中的技术实现了这一目标，该技术使它能够选择坐标作为特定方程解的点。Bubeck说，该模型已生成了一条很长的思维链，使得得出这一答案的提示是一个关于Erdos的猜想是真还是假的开放式问题，并不是一个证明他是错误的明确要求。OpenAI的数学家Mehtaab Swahney说：“看到这个模型像人类一样真正通过推理来解决问题，着实令人惊叹。”
这一推理过程包含在一份长达125页的文件中，但OpenAI尚未完全公布这份文件。此外，该公司也未透露其模型的具体名称。Bubeck表示，这是一个实验性的通用推理模型，并非专为解决数学问题而设计，并且它能够根据一个提示自主完成所有工作，即对Erdos问题给出一个机器重写的表述。
Bubeck表示，这种做法与利用AI解决数学问题的“编排”方法截然不同。在“编排”方法中，研究人员会让大语言模型（LLM）通过不断迭代的方式纠正自身的错误，从而找到问题的解决方案。
相比之下，OpenAI系统给出的答案不会因提示语的表述方式不同而有太大差异。到目前为止，一些针对数学问题的最佳AI解决方案都需要大量试错，而提示语的使用已成为一门艺术。“如今，你基本可以任何你想要的方式提出问题，而模型都会正确理解这些问题。”Bubeck说。
Litt表示，由AI生成的解决方案所采用的算法来自代数数论这一事实表明，AI模型正在超越专业化“孤岛”的局限，实现更广泛的应用。他补充道，没有人能够像LLM那样全面掌握数学领域的文献内容。
“我们所有人都曾预料有朝一日会看到这样的情况，但没想到会这么快。”OpenAI的数学家Mark Sellke说，“这与一个月前我们习以为常的情况相比，是一个巨大的飞跃。”（李木子）
来源：《中国科学报》 (2026-05-25 第2版 国际)