There are an infinite number of prime numbers
We will prove that there are an infinite number of prime numbers.
Let's presume the contrary:
There are only a finite number of prime numbers: p1, p2 ... pn. (A)
Then let m be:
m = p1p2 ... pn+1
Let d be the smallest divisor of m which is different than 1. Any divisor of d is also a divisor of m so d hasn't any divisors that are smaller then d and are different the d. So d is prime.
Then d is one of the pi, with i from 1 to n. This is not possible because d is a divisor of m but pi is not a divisor of m. So we have a contradiction it means that (A) is false.
There are an infinite number of prime numbers.