连续合数定理

[(ッ)]

证明或者证伪：对于任意一个正整数k，可以找到至少k个连续正整数，全部都是合数

[YL7983]

你为啥要问这个问题？

[(ッ)]

你为啥要问这个问题？

说明质数之间的gap可以任意大

[TheMatrix]

说明质数之间的gap可以任意大

嗯。据说最近陶哲轩证明了一个什么和质数合数有关的东西。

[YL7983]

Lol,网上随便搜一搜。就有答案！

[TheMatrix]

Lol,网上随便搜一搜。就有答案！

有答案在这里写一下也是有益的。

[YL7983]

（k+1)!+j, j=2,3,…,k+1.
我严重怀疑你骗取我的点击率！

[TheMatrix]

（k+1)!+j, j=2,3,…,k+1.
我严重怀疑你骗取我的点击率！

哦对呀。

[(ッ)]

（k+1)!+j, j=2,3,…,k+1.
我严重怀疑你骗取我的点击率！

艹，你这就是不讲武德

[randomatrices]

为什么不讲武德？在有些基础数论书书上会一句话提到啊。

艹，你这就是不讲武德

[xiaoju]

素数定理都被证明了还有人研究这个？

证明或者证伪：对于任意一个正整数k，可以找到至少k个连续正整数，全部都是合数

[san721]

证明或者证伪：对于任意一个正整数k，可以找到至少k个连续正整数，全部都是合数

A more interesting question is to ask: How small can n be such that (n, n+k) contains no prime. By Prime Number Theorem, n with size about (1+o(1))e^k is relatively trivial. In 1930's, it was proved that n=o(e^k). There have been a lot of work on this, with the most recent (also significant) progress being the work by Ford, Green, Konyagin, Tao, and Maynard. It is conjectured that the minimum n should be about (1+o(1))e^{\sqrt{k}}.

[TheMatrix]

A more interesting question is to ask: How small can n be such that (n, n+k) contains no prime. By Prime Number Theorem, n with size about (1+o(1))e^k is relatively trivial. In 1930's, it was proved that n=o(e^k). There have been a lot of work on this, with the most recent (also significant) progress being the work by Ford, Green, Konyagin, Tao, and Maynard. It is conjectured that the minimum n should be about (1+o(1))e^{\sqrt{k}}.

不错。

感觉一个问题，能以各种方式扩展。

比如一辆火车，走第一站用了20分钟，下一站25分钟，再下一战......，最后问：走了几站。

[san721]

不错。

感觉一个问题，能以各种方式扩展。

比如一辆火车，走第一站用了20分钟，下一站25分钟，再下一战......，最后问：走了几站。

Hahaha, lmao.

[YWY]

不错。

感觉一个问题，能以各种方式扩展。

比如一辆火车，走第一站用了20分钟，下一站25分钟，再下一战......，最后问：走了几站。

最后的问题，类似于问“第n站最快用多长时间”