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背景：我是一名程序员，不是数学家，但这个故事让我感到相当有趣：<a href="https://news.ycombinator.com/item?id=47903126">https://news.ycombinator.com/item?id=47903126</a>我想知道通过随机选择一个开放问题并将其抛给大型语言模型（LLMs），我能得到多远的结果。声明：我对这个问题的具体含义毫无头绪，更不用说输出结果是否正确了。我的兴趣纯粹是为了测试各种模型的能力、好奇心和娱乐。问题：<a href="https://www.erdosproblems.com/691" rel="nofollow">https://www.erdosproblems.com/691</a><pre><code> 设 \( A \subseteq \mathbb{N} \)，令 \( M_A=\{ n \geq 1 : a\mid n\textrm{ 对于某个 }a\in A\} \) 为 \( A \) 的倍数集合。找出 \( A \) 的必要且充分条件，使得 \( M_A \) 的密度为 1。 </code></pre> 我的方法：我在专家模式下使用 DeepSeek，使用与链接的 HN 提交相同的提示。它思考了很长时间，但我在后台做其他事情，所以没有准确计时。我在大约 60 分钟内按了两次“继续”。输出显示它思考了大约 46 分钟。一旦生成了证明，我请 Opus 4.7 对其进行审查，然后将审查结果输入 DeepSeek，DeepSeek 进行了编辑、修正和完善。这种来回的过程持续到 Opus 4.7 感到相当满意。此时，我引入了 Gemini 3.1 Pro Preview，它提出了 Opus 忽略的问题。Opus 认可了反馈，然后我将其反馈放入 DeepSeek 进行最后一轮。基本上，Opus 所说的 DeepSeek 生成的是“对 D[avenport]-E[rdos] 推论的清晰阐述”，而不是一个新结果。很可能这个结果已经被人们所知（在这个阶段 DeepSeek 不被允许使用互联网），甚至可能是错误的。用“简单”的话来说：<pre><code> 这个论证实际上证明了每个自然数集合 \( A \) 的一个更强的事实：集合 \( M_A \) 的上密度等于从 \( A \) 的有限子集可以得到的最大下密度，这也等于 \( M_A \) 的下密度。当上密度为 1 时，它迫使下密度也为 1，因此自然（普通）密度自动存在并等于 1，而无需任何额外条件。证明中唯一非基础的部分是 Davenport–Erdős 定理；其他部分都很简单。 </code></pre> 无论如何，这是我的收获：- 这些新模型似乎特别强大，尤其是在相互结合使用时，即使是相对简单的提示- 我对 DeepSeek 感到相当印象深刻。我打算审查它的编码能力，甚至可能完全从 Anthropic 切换过来- 这确实是一个有趣的练习，即使我不知道其中是否有任何正确或有用的内容其他一些观察：- Opus 在审查 DeepSeek 的输出时非常迅速，几乎是几秒钟- Gemini 在弄清楚“Erdos 691”指的是什么时遇到了麻烦- ChatGPT 的免费版本生成的输出大多无用。我没有包括它。聊天链接如下：<a href="https://chat.deepseek.com/share/hpguvrhcxn226bi3hn" rel="nofollow">https://chat.deepseek.com/share/hpguvrhcxn226bi3hn</a><a href="https://claude.ai/share/4f3ccad1-d862-4e37-8333-8a1ebd84b38f" rel="nofollow">https://claude.ai/share/4f3ccad1-d862-4e37-8333-8a1ebd84b38f</a><a href="https://aistudio.google.com/app/prompts?state=%7B%22ids%22:%5B%221cRWKS3ngW_nqfSn3W-Kq_bWlk8FWXmDw%22%5D,%22action%22:%22open%22,%22userId%22:%22100878499144503719961%22,%22resourceKeys%22:%7B%7D%7D&usp=sharing" rel="nofollow">https://aistudio.google.com/app/prompts?state=%7B%22ids%22:%...</a>

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Background: I am a coder, not a mathematician, but I was quite entertained by this story:<a href="https://news.ycombinator.com/item?id=47903126">https://news.ycombinator.com/item?id=47903126</a>I wondered how far I could get by just choosing a random open problem and throwing it at LLMs.Disclosure: I have no idea what the problem even refers to, let alone whether or not the output is even remotely correct. My interest is purely for testing capabilities of various models, curiousity, and entertainment.Problem: <a href="https://www.erdosproblems.com/691" rel="nofollow">https://www.erdosproblems.com/691</a><pre><code> Given A\subseteq \mathbb{N} let M_A=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\} be the set of multiples of A. Find a necessary and sufficient condition on A for M_A to have density 1. </code></pre> My approach: I used DeepSeek in Expert mode, using the same prompt as in the linked HN submission. It thought for a very long time, but I was doing other things in the background so I didn't really time it. I pressed "Continue" twice over the space of maybe 60mins. The output says it thought for about 46mins.Once it generated a proof, I asked Opus 4.7 to review it, and then entered the review into DeepSeek which made edits, corrections and refinements. This back-and-forth continued till Opus 4.7 was reasonably happy. At that point, I called in Gemini 3.1 Pro Preview, which raised issues which Opus missed. Opus acknowledged the feedback, and then I placed its feedback into Deepseek for a final round. Essentially, what Opus says Deepseek generated was a "clean exposition of a D[avenport]-E[rdos] corollary", not a new result. In all likelihood this result may already be known (Deepseek was not allowed to use the internet for this phase), or even wrong.In "simple" terms:<pre><code> The argument actually proves a stronger fact for every set \( A \) of natural numbers: The upper density of the set \( M_A \) equals the largest possible lower density you can get from finite subsets of \( A \), and that also equals the lower density of \( M_A \). When the upper density is 1, it forces the lower density to also be 1, so the natural (ordinary) density exists and equals 1 automatically, without needing any extra conditions. The only non-basic part of the proof is the Davenport–Erdős theorem; everything else is simple. </code></pre> In any case, these were my takeaways:- These new models seem to be surprisingly capable especially when used to in conjunction with each other, even with fairly simple prompts- I am quite impressed by Deepseek. I'm going to review its coding ability, and may even switch completely from Anthropic- This was a genuinely interesting exercise, even if I have no idea if any of it is correct or usefulSome other observations:- Opus was really fast at reviewing Deepseek's output. Literally seconds- Gemini had trouble figuring out what "Erdos 691" referred to- The free version of ChatGPT of generated mostly useless output. I didn't include it.Chat links below:<a href="https://chat.deepseek.com/share/hpguvrhcxn226bi3hn" rel="nofollow">https://chat.deepseek.com/share/hpguvrhcxn226bi3hn</a><a href="https://claude.ai/share/4f3ccad1-d862-4e37-8333-8a1ebd84b38f" rel="nofollow">https://claude.ai/share/4f3ccad1-d862-4e37-8333-8a1ebd84b38f</a><a href="https://aistudio.google.com/app/prompts?state=%7B%22ids%22:%5B%221cRWKS3ngW_nqfSn3W-Kq_bWlk8FWXmDw%22%5D,%22action%22:%22open%22,%22userId%22:%22100878499144503719961%22,%22resourceKeys%22:%7B%7D%7D&usp=sharing" rel="nofollow">https://aistudio.google.com/app/prompts?state=%7B%22ids%22:%...</a>

展示HN：另一个关于厄尔多斯问题和大型语言模型的实验