Show HN：关于一个埃尔德什问题的 LLM 实验
1 分•作者: ilitirit•3 个月前
背景：我是一名程序员，不是数学家，但这个故事让我很感兴趣：
<p><a href="https://news.ycombinator.com/item?id=47903126">https://news.ycombinator.com/item?id=47903126</a></p>
我想知道，如果我只是选择一个随机的未解决问题，然后把它交给大型语言模型，我能走多远。
声明：我根本不知道这个问题指的是什么，更不用说输出是否正确了。我感兴趣的纯粹是测试各种模型的能力、好奇心和娱乐性。
问题：<a href="https://www.erdosproblems.com/691" rel="nofollow">https://www.erdosproblems.com/691</a>
<pre><code> 给定 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。
求 A 的一个充要条件，使得 M_A 的密度为 1。
</code></pre>
我的方法：
我使用了 DeepSeek 的专家模式，使用了与链接的 HN 提交中相同的提示。它思考了很长时间，但我在后台做其他事情，所以实际上没有计时。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。
一旦它生成了一个证明，我让 Opus 4.7 审查它，然后将审查结果输入 DeepSeek，DeepSeek 进行编辑、更正和完善。这种反复进行的过程一直持续到 Opus 4.7 相当满意为止。在那一点上，我调用了 Gemini 3.1 Pro Preview，它提出了 Opus 遗漏的问题。Opus 承认了反馈，然后我将它的反馈输入 Deepseek 进行最后一轮。本质上，Opus 说 Deepseek 生成的是“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为人所知（Deepseek 在这个阶段不允许使用互联网），甚至可能是错的。
用“简单”的术语来说：
<pre><code> 该论证实际上证明了对于每个自然数集合 \( A \) 来说，一个更强的事实：
集合 \( M_A \) 的上密度等于你可以从 \( A \) 的有限子集中获得的最大可能下密度，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，不需要任何额外的条件。
证明中唯一非基本的部分是 Davenport–Erdős 定理；其他一切都很简单。
</code></pre>
无论如何，这是我的收获：
* 这些新模型似乎出奇地强大，特别是当它们相互配合使用时，即使使用相当简单的提示也是如此
* 我对 Deepseek 印象深刻。我将审查它的编码能力，甚至可能完全从 Anthropic 切换
* 这是一个真正有趣的练习，即使我不知道它是否正确或有用
其他一些观察：
* Opus 在审查 Deepseek 的输出方面非常快。字面意思就是几秒钟
* Gemini 很难弄清楚“Erdos 691”指的是什么
* 免费版的 ChatGPT 生成了大部分无用的输出。我没有包括它。
聊天链接如下：
<p><a href="https://chat.deepseek.com/share/hpguvrhcxn226bi3hn" rel="nofollow">https://chat.deepseek.com/share/hpguvrhcxn226bi3hn</a></p>
<p><a href="https://claude.ai/share/4f3ccad1-d862-4e37-8333-8a1ebd84b38f" rel="nofollow">https://claude.ai/share/4f3ccad1-d862-4e37-8333-8a1ebd84b38f</a></p>
<p><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></p>
查看原文
Background: I am a coder, not a mathematician, but I was quite entertained by this story:<p><a href="https://news.ycombinator.com/item?id=47903126">https://news.ycombinator.com/item?id=47903126</a><p>I wondered how far I could get by just choosing a random open problem and throwing it at LLMs.<p>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.<p>Problem: <a href="https://www.erdosproblems.com/691" rel="nofollow">https://www.erdosproblems.com/691</a><p><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.<p>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.<p>In "simple" terms:<p><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:<p>- These new models seem to be surprisingly capable especially when used to in conjunction with each other,
even with fairly simple prompts<p>- I am quite impressed by Deepseek. I'm going to review its coding ability, and may even switch completely from Anthropic<p>- This was a genuinely interesting exercise, even if I have no idea if any of it is correct or useful<p>Some other observations:<p>- Opus was really fast at reviewing Deepseek's output. Literally seconds<p>- Gemini had trouble figuring out what "Erdos 691" referred to<p>- The free version of ChatGPT of generated mostly useless output. I didn't include it.<p>Chat links below:<p><a href="https://chat.deepseek.com/share/hpguvrhcxn226bi3hn" rel="nofollow">https://chat.deepseek.com/share/hpguvrhcxn226bi3hn</a><p><a href="https://claude.ai/share/4f3ccad1-d862-4e37-8333-8a1ebd84b38f" rel="nofollow">https://claude.ai/share/4f3ccad1-d862-4e37-8333-8a1ebd84b38f</a><p><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>