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「Deterministic methods to find primes」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

I’ve uploaded to the arXiv the polymath research paper “D eterministic methods to find primes “, which is the outcome of the Polymath4 collaborative mathematics project, and has been submitted to Mathematics of Computation .

The objective of this paper was to find fast deterministic algorithms to solve the following problem:

已知结果和反例

Given a (large) integer x, find a prime p larger than x.

Thanks to the AKS algorithm , a number of size O(x) can be deterministically tested for primality in time . By Bertrand’s postulate , there is always at least one prime between x and 2x; by testing each one of these integers in turn for primality, one can thus obtain a deterministic algorithm to find primes in time .

证明或构造的主线

But one should be able to do much better. For comparison, if probabilistic algorithms are allowed, then by randomly selecting integers between x and 2x to test for primality, it is easy to see from the prime number theorem that one will succeed in obtaining a prime with high probability in time . However, after some effort we were not able to “derandomise” this algorithm to create any reasonable deterministic counterpart. Nevertheless, we conjecture that a deterministic algor

Currently, the best known deterministic algorithm is due to Lagarias and Odlyzko , and has a run time of . Roughly speaking, it is based on the ability to compute the prime counting function in time ; once one has this function, one can detect which intervals contain primes or not, and then starting from Bertrand’s postulate and performing a binary search one can then locate a prime. The Lagarias-Odlyzko argument is based on approximating by a certain integral of the Riemann

阅读时建议盯住的点

We conjecture that one should be able to compute in faster time, and in particular in time for some . Unfortunately, we were not able to achieve this; however, we do have a non-trivial method to compute the parity of in such a time; a bit more generally (and oversimplifying a little bit), we can compute various projections of the prime polynomial modulo some small polynomials g. This seems close to being able to achieve the goal of detecting whether primes exist in a given in

Roughly speaking, the idea to compute the parity of is as follows. The first observation is that, for square-free n, the number of divisors of n is equal to 2 when n is a prime, and a multiple of 4 otherwise. So to compute the parity of , it suffices to compute modulo 4 (or more precisely, the restriction of this sum to squarefree n).

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