Sums of Two Squares

During the next four lectures, we will study binary quadratic forms. A simple example of a binary quadratic form that will occupy us today is

$$x^2 + y^2.$$

A typical question that one asks about a quadratic form is which integers does it represent. ``Are there integers $x$ and $y$ so that $x^2 + y^2 = 389$? So that $x^2 + y^2 = 2001$?''