1 Prime numbers

Suppose for the sake of contradiction that there are only finitely many prime numbers. Let them be listed as $p_{1}$, $\dots$, $p_{n}$, and set $P=p_{1}\dotsm p_{n}+1$. Since $P>1$, by 事実1.1, $P$ must have at least one prime factor, which we will call $q$. Since $q$ is a prime number, it must be either one of the primes $p_{1}$, $\dots$, $p_{n}$. However, this contradicts to the fact that $q$ divides $P$ ¹¹ 1 It follows from the choice of $q$.. Therefore, there are infinitely many prime numbers. ∎