陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Decomposing a factorial into large factors (second version)」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Boris Alexeev , Evan Conway, Matthieu Rosenfeld , Andrew Sutherland , Markus Uhr, Kevin Ventullo , and I have uploaded to the arXiv a second version of our paper “ Decomposing a factorial into large factors “. This is a completely rewritten and expanded version of a previous paper of the same name . Thanks to many additional theoretical and numerical contributors from the other coauthors, we now have much more precise control on the main quantity studied in this paper, allowi

As discussed in the previous post , denotes the largest integer such that the factorial can be expressed as a product of factors, each of which is at least . Computing is a special case of the bin covering problem , which is known to be NP-hard in general; and prior to our work, was only computed for ; we have been able to compute for all . In fact, we can get surprisingly sharp upper and lower bounds on for much larger , with a precise asymptotic

已知结果和反例

for an explicit constant , which we conjecture to be improvable to

for an explicit constant : … For instance, we can demonstrate numerically that

证明或构造的主线

As a consequence of this precision, we can verify several conjectures of Guy and Selfridge , namely

Guy and Selfridge also claimed that one can establish for all large purely by rearranging factors of and from the standard factorization of , but surprisingly we found that this claim (barely) fails for all :

阅读时建议盯住的点

The accuracy of our bounds comes from several techniques:

To me, the biggest surprise was just how stunningly accurate the linear programming methods were; the very large number of repeated prime factors here actually make this discrete problem behave rather like a continuous one.

值得单独记下的条目

• for all . (In fact we show this is true for , and that this threshold is best possible.)
• Greedy algorithms , in which one allocates the largest prime factors of first and then moves to smaller primes, provide quickly computable, though suboptimal, lower bounds on for small, medium, and moderately large values;
• Linear programming and integer programming methods provides extremely accurate upper and lower bounds on for small and medium values of ;
• Rearrangement methods can be analyzed asymptotically via linear programming, and work well for large ; and

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源