Prove there is infinite prime

\[
\mbox{There is infinite prime}
\]
Assume there is finite number of primes from $p_1 \dots p_k$

and let $n = p_1 p_2 \dots p_k + 1$

$\therefore n - p_1 p_2 \dots p_k = 1$

$P$ is not a prime

let $q \mid P$ but $ q \nmid p_1 p_2 \dots p_k$ otherwise $ \frac{P}{q} - \frac{p_1 p_2 \dots p_k}{q} = \frac{1}{q}$ which is impossible

since left side is integer and right side is non integer

Therefore $P$ must be a prime and has not prime factor from $p_1 p_2 \dots p_k$

Hence, there are infinite primes

and let $n = p_1 p_2 \dots p_k + 1$

$\therefore n - p_1 p_2 \dots p_k = 1$

$P$ is not a prime

let $q \mid P$ but $ q \nmid p_1 p_2 \dots p_k$ otherwise $ \frac{P}{q} - \frac{p_1 p_2 \dots p_k}{q} = \frac{1}{q}$ which is impossible

since left side is integer and right side is non integer

Therefore $P$ must be a prime and has not prime factor from $p_1 p_2 \dots p_k$

Hence, there are infinite primes