There Are Infinitely Many Primes

When one asks the question, What do mathematicians do? the immediate answer given by most is that they prove theorems. In this paper we will consider proofs of Euclid's theorem that there are infinitely many primes. Euclid proved this theorem in book IX of his Elements [Eu] which is easily the most influential book in mathematics ever written. It is not well-known that Euclid was a substantial number-theorist.