Proof of Infinitely Many Primes

Already 300 BC Euclid was able to proof in his Elements that there are infinitely many primes. We can proof it as follows:

Let’s assume only limited number of primes $p_1, \dots, p_n$ exist. Then, there is a number $m$ which can be written as the product of these prime numbers.

$$m=p_1\cdot p_2\cdot\ \ldots\ \cdot p_n$$

If we add the number 1 to this number we have two options:

Either $m+1$ is a prime number. This would contradict our assumption and therefore our assumption would be wrong and there must be an infinitely number of primes.

Or $m+1$ is no prime number. Then, there must be some $p_i$ which divides $m$ and $m+1$. That would mean (see here) that $p_i$ also divides the difference $(m+1)-m=1$. That would mean that $p_i$ must divide $1$ which is not possible as $p_i$ is prime and therefore greater than $1$. Again, our assumption must be wrong and there must be an infinitely number of primes.