Sums of Four Squares

The main results to be presented here deal with representations as a sum of four squares. However, it is useful for purposes of exposition to consider the corresponding theorems for sums of two squares. Since these results are so familiar, and part of elementary courses, it may seem that these propositions are belaboring the obvious. However, there is a slight difference in emphasis from the usual treatment that will be useful in describing the generalization. There are two types of questions to be considered: algorithmic — how can one compute representations of a number as a sum of two or four (or possibly some other number) of squares? — and enumerative — is there a structure on the set of representations that allows their number to be determined in an elementary manner?