It is an old result of Euclid that there are infinitely many primes. One way to refine this is to ask how the primes are distributed in arithmetic progressions. For example, there are three arithmetic progressions with common difference 3:

The proof of results like this is quite different from Euclid's relatively simple proof; it involves analytic techniques. In this talk we will illustrate the ideas for arithmetic progressions having common difference 3 (as above) and 5. No background beyond second semester calculus will be required.