Sums of squares is an important topic in number theory; it represents a melting pot for various methods, extending from elementary ideas to advanced analytic and algebraic techniques. This book presents many facets of ideas developed in the past to solve problems connected with the representation of integers as sums of squares. Starting with elementary methods that have been used since the field was young and ranging to Liouville’s contribution, the book continues through the theory of modular forms and the analytical methods used to count the representations. An interesting extension (in comparison with other books on the subject) is the two chapters devoted to arithmetic progressions and applications to real life problems. The first one deals with the theorem of van der Waerden, Roth and Szemerédi (probably the first book that gives a proof of this deep result) and the second one covers connections to factorization and applications in the areas of microwave radiation, diamond cutting and cryptanalysis.

The book is written in a very fresh style and it is essentially self-contained (e.g. the Riemann-Roch theorem is used but not proved) and it includes about a hundred interesting exercises of various levels to test the reader’s understanding of the text. The book can be recommended to those interested in the development of ideas in the subject and their context.

Reviewer:

spor