How many prime numbers exist? (Euclid's proof)

How many prime numbers are there? Will we ever run out? Euclid's proof, elegant and beautiful in how simply and efficiently it deals with such a seemingly enormous problem, settled it once and for all. That's one of the coolest things about maths - once you prove something, it's been proved forever.
This video comes from a series of lessons about cryptography (prime numbers are hugely important to modern encryption algorithms).