Primes are the building blocks of the set of numbers. Every number can be broken down into a product of prime numbers in a unique way. This is a fact that seems obvious but does require some effort to verify. In this chapter we will explore a number of statements about the collection of prime numbers. Some of these have relatively simple proofs, some of them have very difficult proofs, and some of these are merely conjectures that have not yet been penetrated by any human mind. We begin with an ancient method for quickly determining the primes less than some fixed number.