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「“New equidistribution estimates of Zhang type, and bounded gaps between primes” – and a re」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

The first purpose is to announce the uploading of the paper “ New equidistribution estimates of Zhang type, and bounded gaps between primes ” by D.H.J. Polymath, which is the main output of the Polymath8a project on bounded gaps between primes , to the arXiv, and to describe the main results of this paper below the fold.

The second purpose is to roll over the previous thread on all remaining Polymath8a-related matters (e.g. updates on the submission status of the paper) to a fresh thread. (Discussion of the ongoing Polymath8b project is however being kept on a separate thread , to try to reduce confusion.)

已知结果和反例

The final purpose of this post is to coordinate the writing of a retrospective article on the Polymath8 experience, which has been solicited for the Newsletter of the European Mathematical Society . I suppose that this could encompass both the Polymath8a and Polymath8b projects, even though the second one is still ongoing (but I think 下面会 soon be entering the endgame there). I think there would be two main purposes of such a retrospective article. The first one would be to te

With these two objectives in mind, I propose a format for the retrospective article consisting of a brief introduction to the polymath concept in general and the polymath8 project in particular, followed by a collection of essentially independent contributions by different participants on their own experiences and thoughts. Finally we could have a conclusion section in which we make some general remarks on the polymath project (such as the remarks above). I’ve started a dropb

证明或构造的主线

As for deadlines, the EMS Newsletter would like a submitted article by mid-April in order to make the June issue, but in the worst case, it will just be held over until the issue after that.

where denotes the prime. Thus for instance the notorious twin prime conjecture is equivalent to the claim that . However, even establishing the finite nature of unconditionally was an open problem until the celebrated work of Zhang last year, who established the bound

阅读时建议盯住的点

Zhang’s argument, which built upon earlier work of Goldston, Pintz, and Yildirim , can be summarised as follows. For any natural number , define an admissible -tuple to be a tuple of increasing integers, which avoids at least one residue class modulo for each prime . For instance, is an admissible -tuple, but is not. The Hardy-Littlewood prime tuples conjecture asserts that if is an admissible -tuple, then there exists infinitely many such that are simultaneously prime. This

Zhang was able to obtain for , and then took the first primes larger than to be the admissible -tuple, observing that this tuple had diameter at most . (Actually, it has diameter , as observed by Trudgian .) The earliest phase of the Polymath8a project consisted of using increasingly sophisticated methods to search for narrow admissible tuples of a given cardinality; in the case of this particular , we were able to find an admissible tuple whose diameter was . On the other ha

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