Here is a proof based on Euclid (this is not the way Euclid proved the theorem):
Prove that there are infinitely many primes.
We assume that there are a finite number of primes. Create the following number:
`P=p_1*p_2*...*p_n+1` where `p_i` is a prime number, and n the assumed number of primes. Thus this is the product of every known prime plus 1.
By the fundamental theorem of arithmetic `P` can be written uniquely as the product of primes. But none of the known primes divides `P` as they all leave a remainder of 1. Thus either `P` itself is prime, or there is a new prime that divides `P` .
This contradicts our assumption that there is a finite group of primes; there fore the number of primes is infinite.