# There are an infinite number of prime numbers

Submitted by Isoscel on Tue, 09/25/2007 - 22:30.

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: p_{1}, p_{2} ... p_{n}. (A)

Then let m be:

m = p_{1}p_{2} ... p_{n}+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 p_{i}, with i from 1 to n. This is not possible because d is a divisor of m but p_{i} is not a divisor of m. So we have a contradiction it means that (A) is false.

In conclusion:

There are an infinite number of prime numbers.