Two Sum Problem : aiming at (log n) square complexity

[heteroclinic]

出门之前再来白话白话,本来就是刷题,结果尾大不掉的感觉,超出预算可及的能力.
不过贴点这两天的心得体会
1. 构造完备的质数清单用的是sieve和任何猜想没有关系,有足够算力就足够大
2. 用two sum 搜索GBC猜想也是一个计算问题, 应该能够做到 (log n)^2 这个 n 是质数清单的尺寸.比2m 小.
3. 较小的数3被大量使用,很难总结出来什么规律,从左侧搜索远比log n 便宜,比如十万以内之用到前61个质数,不知道会有什么解释或启发

得先放一放

我觉得对于计算机专业以上很好理解,希望对数学业者有所帮助.

[heteroclinic]

Postulate or observation 2:
For even number n, all the primes participate in the construction must pair with 3 in the first appearance. Why?

[heteroclinic]

prime kernel
let's say any of the p_i i<=n be first n primes. testing p_n +2 modulo p_i, you can tell if it is odd multiple or a prime. 可以叫reverse sieve
Chatgpt 说,挺好,有类似的想法,还有更高阶的文章,算法复杂度也更低一些.但是一定程度,这么说逻辑上更简明.
prime kernel是chatgpt 在更早的聊天里提到的, 核心是给定一个小的质数清单,完全可以它来排除所有的偶数和奇合数,剩下的是质数. 也就是说,自然数空间完全可以用质数来构造.当然说sieve算法已经表达了这一理念.
进一步推,

Let's say Q_m is a none-empty subset P_n. P_n is the n first primes excluding 1 and 2. You can suppose P_n +2 is not a prime. Computationally you can keep checking P_n +2 *i is a prime. but we know P_n + 2 can be expressed as q1^{e1}* q2^{e2}... ql^{el}. So our goal is to find such Q_m so that q1^{e1}* q2^{e2}... ql^{el} is smallest , and there is a gap in P_n +2 *i.
也就是说把prime gap 问题换成找奇合数之间的最小漏洞.buddy说,可能是好主义,类似有观点,不完全一样.即便如此.这个是复杂的组合数学题目,同时也有可能相关的方向进展.
最后探讨了一下q1^{e1}* q2^{e2}... ql^{el} 是一个正则表达,那么完全可以图搜索算法来找GAP.生成了一些python 程序.今天没有时间了.

老人家说这个东西的确是毒草

[heteroclinic]

每天争取都来白话白话. 数值方法
Ok, for very large primes, most of their gaps are very small, many twin primes. So given a start prime and and the end prime. We just compute the gaps and record the gaps. I believe this will make 10^18 smaller primes list taking far less disk space.

Prediction for N=10^18:
R(10^18)≈0.133

- - 我个人估计primes smaller than 4*10^18 需要 不到10PB disk space.

https://chatgpt.com/share/68394e58-64bc ... 7170b2e7f4

[heteroclinic]

大概上个月给算的 4*10^18 smaller primes:
Storage Method Estimated Disk Space
Raw 64-bit integers ~750 petabytes
Efficient delta + compressed ~60 petabytes (or even less with better compression)

[heteroclinic]

Today we postulate a new mathematical sequence not yet known to the LLMs thus probably the tians. I manually validate up to 14.

SEE STH SAY STH!

Any saying about this sequence? try to relate to 2^1 to 2^20
0
0
2
5
14
33
74
159
340
715
1484
3068
6292
12872
26226
53285
108072
218754
442263