We examine methods to produce triples of integers which are the sides of a right triangle (i.e., (3,4,5), (5,12,13), or (8,15,17)). Multiplication of complex numbers will make an appearance, the first example of the interplay between algebra and geometry. We will then learn about elliptic curves, which provide a similar, but much more intricate, synthesis of algebra and geometry. Elliptic curves are a current area of intense research in mathematics and computer science, playing a central role in modern cryptology and in the recent proof of Fermat's Last Theorem.