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「A truncated elementary Selberg sieve of Pintz」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

This post is a continuation of the previous post on sieve theory , which is an ongoing part of the Polymath8 project . As the previous post was getting somewhat full, we are rolling the thread over to the current post.

在这类讨论里 下面会 record a new truncation of the elementary Selberg sieve discussed in this previous post (and also analysed in the context of bounded prime gaps by Graham-Goldston-Pintz-Yildirim and Motohashi-Pintz ) that was recently worked out by Janos Pintz, who has kindly given permission to share this new idea with the Polymath8 project. This new sieve decouples the parameter that was present in our previous analysis of Zhang’s argument into two parameters, a quantity that use

已知结果和反例

To describe this new truncation we need to review some notation. As in all previous posts (in particular, the first post in this series ), we have an asymptotic parameter going off to infinity, and all quantities here are implicitly understood to be allowed to depend on (or to range in a set that depends on ) unless they are explicitly declared to be fixed . We use the usual asymptotic notation relative to this parameter . To be able to ignore local factors (such as the singu

For any fixed natural number , define an admissible -tuple to be a fixed tuple of distinct integers which avoids at least one residue class modulo for each prime . Our objective is to obtain the following conjecture for as small a value of the parameter as possible:

证明或构造的主线

Conjecture 1 ( ) Let be a fixed admissible -tuple. Then there exist infinitely many translates of that contain at least two primes.

The twin prime conjecture asserts that holds for as small as , but currently we are only able to establish this result for (see this comment ). However, with the new truncated sieve of Pintz described in this post, we expect to be able to lower this threshold somewhat.

阅读时建议盯住的点

In previous posts, we deduced from a technical variant of the Elliot-Halberstam conjecture for certain choices of parameters , . We will use the following formulation of :

Conjecture 2 ( ) Let be a fixed -tuple (not necessarily admissible) for some fixed , and let be a primitive residue class. Then

值得单独记下的条目

• (i) could be divisible by a prime with .
• (ii) could be divisible by a prime with .

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