The oldest known proof that there are infinite number of primes is the one that is due to Euclid (around 300 BC). It is probably the most accessible. So I start the blog by stating this proof.
Theorem. There are infinitely many prime numbers.
The strategy is to show that given a finite list of prime numbers , we can always find another prime number not on the list.
Let . Let be a prime divisor of . If is on the list , then would divide both and . This means that would divide 1 (the difference of and ). This is impossible. So must a prime number not on the original list. Hence there must be infinitely many prime numbers.