How many prime numbers exist?

There are an infinite amount of prime numbers. The first to have proven this was Euclid. Here are the general lines of his proof:

# Suppose there are n prime numbers overall. # Let N be a common multiple of all these primes.

# Is N+1 prime? If it is, we have found a new prime.

# Suppose N+1 is not prime. Thus, there exists a Prime number p which divides N+1 evenly. If p is one of the primes dividing N, it also divides 1 evenly, which is impossible. Thus, p is not one of the n primes, and we found a new prime.