The Primes: Infinitude, Density and Substance

The title of this section is surely, along with the uniqueness of factorization, the most basic and important fact in number theory. The first recorded proof was by Euclid, and we gave it at the beginning of the course. There have since been (very!) many other proofs, many of which have their own merits and drawbacks. It is entirely natural to look for further proofs: in terms of the arithmetical function π(n) which counts the number of primes p ≤ n, Euclid’s proof gives that