# 試證明質數有無限多個

Give an integer $n\geq 3$, Claim there exists a prime number $p$ such that $n < p < n!$

Consider $a = n!-1$, then there must be some prime number $p < n!$ such that $p \mid a$.

If $p \leq n$, then $p \mid n!$, thus we have $p \mid (n!-a)$, that is $p \mid 1$, contradiction. Hence $n < p < n!$.

Take ${a_i \mid a_1 = 3, a_{i+1} = (a_i)!, \forall i\geq 1}$, given positive integer $i$, there exists some $p_i$ s.t. $a_i < p_i < a_{i+1} = (a_i)!$, and ${p_i}$ is strictly increasing, then we are done.

$$\tag{*A}\prod\limits_{i=1}^{n}\sum\limits_{j=0}^{\infty}\frac{1}{p_i^j} = \prod\limits_{i=1}^{n}\frac{1}{1-1/p_i}$$

$$\tag{*B}\prod\limits_{i=1}^{n}\sum\limits_{k=1}^{\infty}\frac{1}{p_i^j} = \sum\limits_{j=0}^{\infty}\frac{1}{p_1^{r_1}}\times\frac{1}{p_2^{r_2}}\times\cdots\times\frac{1}{p_n^{r_n}}$$

$$\tag{*C}\prod\limits_{i=1}^{n}\sum\limits_{j=0}^{\infty}\frac{1}{p_i^j} = \sum\limits_{k=1}^{\infty}\frac{1}{k}$$

*C 的值卻是發散。於是我們得到矛盾，因此質數有無限多個。