Prime numbers/Infinity/Fact/Proof

Proof

We assume that the set of all prime numbers is finite, say ${}\{p_{1},p_{2},\ldots ,p_{r}\}$ is a complete list of all prime numbers. We consider the number

${}N=p_{1}\cdot p_{2}\cdot p_{3}\cdots p_{r}+1\,.$

This number can not be divided by any of the prime numbers ${}p_{i}$, since the reminder of ${}N$ by division through ${}p_{i}$ is always ${}1$. Hence, the prime factors of ${}N$, which exist due to

are not contained in the given set. This is a contradiction.

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