ABSTRACT
The problem of representing a positive integer as a sum of s squares has a long and interesting history, partly recounted in Dickson’s monumental history of number theory [24], volume II, pages 225-339. The two best-known results of elementary number theory which have played a significant role in the development of various approaches to the problem are Fermat’s theorem that any prime of the form p = 4m+ 1 is representable in a unique way as a sum of of two squares and Lagrange’s theorem that every positive integer is the sum of at most four squares of integers. As we have seen in Chapter 2, each of these results comes equipped with an explicit formula for the number of possible representations. It has been the goal of many number theorists to obtain similar exact formulas for the number of representations of an integer as the sum of s squares for s > 4. We shall see in this chapter, using the Hecke theory of modular forms, to what extent such a goal is possible.
