elementary number theory – Proving there are infinitely many primes in 5 ways

I need to remember for the test 5 different ways of proving that there are infinitely many primes. Now, I have learned to prove the infinity of primes by Euclid’s theorem, Euler’s theorem, Fermat numbers, Dirichlet, and the last I don’t remember. I will be happy to see how do you suggest to prove these proofs, so I can remember this for the test. It is my first course in number theory, so I only got elementary tools.