How many prime numbers are theree?

Answer 1

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

1. Suppose there are n prime numbers overall.
2. Let N be a common multiple of all these primes.
3. Is N+1 prime? If it is, we have found a new prime.
4. 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.

(see the related link for a list of the first 500 prime numbers)

Answer 2

That's an infinite amount.