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 remark on primality testing and the binary expansion」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 just uploaded to the arXiv the short note “ A remark on primality testing and the binary expansion “, submitted to the Journal of the Australian Mathematical Society . In this note I establish the following result: for any sufficiently large integer n, there exists an n-bit prime p, such that the numbers for are all composite. In particular, if one flips any one of the bits in the binary expansion of the prime p, one obtains a composite number. As a consequence, one obta

Primes p of the form mentioned in the above result are somewhat rare at first; the first some prime is . But in fact, the argument in my note shows that the set of such primes actually has positive relative density inside the set of all primes. (Amusingly, this means that one can apply a theorem of Ben Green and myself and conclude that there are arbitrarily long arithmetic progressions of such primes, although I doubt that there is any particular significance or application

已知结果和反例

The same remark applies to other bases; thus, for instance, there exist infinitely many prime numbers with the property that if one changes any one of the base 10 digits of that number, one obtains a composite number. (Presumably the first such number can be located by computer search, though I did not attempt to do so.) [ Update , Feb 25: see comments.]

Heuristically, a random n-bit integer has roughly a 1/n chance (up to multiplicative constants such as ) of being prime. So amongst the 2n numbers for and a fixed prime p, one expects about O(1) of these numbers to be prime on the average. A more refined calculation (using the Hardy-Littlewood prime tuples conjecture ) suggests that the expected number of primes exceeds 1 (in fact it should be asymptotically , where is the twin prime constant ), which means that the main resu

证明或构造的主线

It is nice for a change to write an elementary paper of six eight pages in length; the only moderately advanced tools I use here are the prime number theorem in arithmetic progressions, and the standard upper bound from the Selberg sieve for any two-dimensional sieving problem (in which one is allowed to eliminate up to two residue classes mod q for every prime q).

[ Update , Feb 25: It has been pointed out to me that the base 2 result was implicitly obtained by Cohen and Selfridge , and explicitly pointed out by Sun , using the covering congruence method of Erdős , so I have rewritten the paper to focus primarily on the case of general bases (to which it is not entirely clear that a set of covering congruences exists). The earlier version (which focused only on the base 2 case, and so is slightly simpler) can be found here .]

阅读时建议盯住的点

[ Update , Feb 26: It should now also be possible to prove the following generalisation: for any base a and any r, there should exist infinitely many primes which become composite whenever one changes one of the digits, and also appends or deletes up to r of digits on either end of the digit expansion. My method comes very close to establishing this result, except for one annoying issue arising from the fact that numbers such as (with j, l = O(1)) can occasionally have a huge

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