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Publisher:

Chapman & Hall/CRC

Publication Date:

2006

Number of Pages:

354

Format:

Hardcover

Series:

Discrete Mathematics and Its Applications 37

Price:

89.95

ISBN:

1-58488-456-8

Category:

Textbook

[Reviewed by , on ]

Darren Glass

05/28/2006

What integers can be written as the sum of two square numbers? Which ones can be written as the sum of three square numbers? Four? And for those numbers that can be written in such a way, how many ways can you do it?

These are questions that pop up in most courses in elementary number theory courses, and some results can be obtained using fairly elementary techniques. But the story does not end there, and these types of questions quickly lead into questions about topics such as Bernoulli numbers, L-functions, modular forms, and arithmetic progressions, many of which require quite a lot of sophisticated machinery to understand the definitions, let alone some of the deep and exciting results. The new book by Carlos J. Moreno and Samuel S. Wagstaff Jr, appropriately entitled *Sums of Squares of Integers* is designed to introduce the reader to this circle of ideas and the connections between the various topics. The prerequisites for the book are all topics that would be familiar to most beginning graduate students — some elementary number theory, some group theory, and complex analysis “at the level of Ahlfors.” Occassionally the authors will refer to more advanced topics, but for the most part the authors succeed in keeping the book quite self-contained.

Assume that a number of the form 2x^{2} can be written as the sum of two squares: that is, m^{2}+ n^{2}= 2x^{2}. Then if follows that the numbers m^{2}, x^{2}, and n^{2} lie in an arithmetic progression. This observation connects the study of sums of squares of integers to the study of arithmetic progressions, and Moreno and Wagstaff use this connection to justify including a very nice chapter in their book on arithmetic progressions. This chapter includes a proof of the theorem of Szemeredi stating that every subset of the positive integers with positive asymptotic density contains arbitrarily long arithmetic progressions which has received quite a bit of attention recently due to the work of Green and Tao on prime numbers in arithmetic progressions.

The approach the authors take throughout their book is one that this reviewer would like to see in more books: they choose a relatively elementary set of questions and approach it from a variety of different angles — combinatorial, analytic, and algebraic — and in doing so they introduce topics from quite a few exciting areas of research and show some of the connections between these approaches. The authors end with a chapter about applications of these results to areas such as microwave radiation, diamond cutting, and cryptography. This is a nice touch, but to be honest I cannot imagine any reader reaching the end and not being convinced that these questions are inherently interesting and lead to some beautiful mathematics.

Darren Glass (dglass@gettysburg.edu) is an Assistant Professor at Gettysburg College.

Introduction

Prerequisites

Outline of Chapters 2 - 8

Elementary Methods

Introduction

Some Lemmas

Two Fundamental Identities

Euler's Recurrence for Sigma(n)

More Identities

Sums of Two Squares

Sums of Four Squares

Still More Identities

Sums of Three Squares

An Alternate Method

Sums of Polygonal Numbers

Exercises

Bernoulli Numbers

Overview

Definition of the Bernoulli Numbers

The Euler-MacLaurin Sum Formula

The Riemann Zeta Function

Signs of Bernoulli Numbers Alternate

The von Staudt-Clausen Theorem

Congruences of Voronoi and Kummer

Irregular Primes

Fractional Parts of Bernoulli Numbers

Exercises

Examples of Modular Forms

Introduction

An Example of Jacobi and Smith

An Example of Ramanujan and Mordell

An Example of Wilton: t (n) Modulo 23

An Example of Hamburger

Exercises

Hecke's Theory of Modular Forms

Introduction

Modular Group ? and its Subgroup ? 0 (N)

Fundamental Domains For ? and ? 0 (N)

Integral Modular Forms

Modular Forms of Type Mk(? 0(N);chi) and Euler-Poincare series

Hecke Operators

Dirichlet Series and Their Functional Equation

The Petersson Inner Product

The Method of Poincare Series

Fourier Coefficients of Poincare Series

A Classical Bound for the Ramanujan t function

The Eichler-Selberg Trace Formula

l-adic Representations and the Ramanujan Conjecture

Exercises

Representation of Numbers as Sums of Squares

Introduction

The Circle Method and Poincare Series

Explicit Formulas for the Singular Series

The Singular Series

Exact Formulas for the Number of Representations

Examples: Quadratic Forms and Sums of Squares

Liouville's Methods and Elliptic Modular Forms

Exercises

Arithmetic Progressions

Introduction

Van der Waerden's Theorem

Roth's Theorem t 3 = 0

Szemeredi's Proof of Roth's Theorem

Bipartite Graphs

Configurations

More Definitions

The Choice of tm

Well-Saturated K-tuples

Szemeredi's Theorem

Arithmetic Progressions of Squares

Exercises

Applications

Factoring Integers

Computing Sums of Two Squares

Computing Sums of Three Squares

Computing Sums of Four Squares

Computing rs(n)

Resonant Cavities

Diamond Cutting

Cryptanalysis of a Signature Scheme

Exercises

References

Index

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