This book is primarily a classical introduction to the subject, with a few modern applications. In the preface, Burton describes how the book's sixteen chapters divide naturally into two parts. The first nine chapters give a detailed introduction to the basics of elementary number theory: Euclidean algorithm, Sieve of Eratosthenes, Chinese remainder theorem, Fermat's and Wilson's theorems, Möbius inversion, Euler's phifunction, and quadratic reciprocity. Chapters 1016 are independent of one another and cover a variety of topics, such as Cryptography, Nonlinear Diophantine Equations, Continued Fractions, and the Riemann zeta function.
The book is full of examples and proofs are carried out in detail. The exercises are plentiful and range from routine to moderately difficult. Burton writes clearly and goes to great pains not to lose the reader during the more intricate arguments; this is most apparent in his treatment of quadratic reciprocity. He builds up the theory slowly through many examples and short theorems. By the time he arrives at Gauss' Lemma (the main tool in his proof of quadratic reciprocity), the reader has become an expert on the various properties of the Legendre symbol.
What makes this book especially enjoyable is the historical background provided for each new topic. These accounts are short (typically 12 pages), frequent, and blend seamlessly with the mathematics. My favorite examples come from chapter 15 — Continued Fractions. Burton begins with a section on Ramanujan's life and his contributions to the subject. Next he develops properties of finite and infinite continued fractions, complete with historical anecdotes. He concludes the chapter with a detailed history and exposition of Pell's equation. His approach works very well; he piques the reader's curiosity with the beauty of Ramanujan's formulæ, then hints of the connection between periodic continued fractions and Pell's equation. This sets the tone for the rest of the chapter and provides clear insight into a subject which can often seem mysterious.
The mathematical content of the last seven chapters is not as deep as that of the first nine, which is to be expected due to their logical independence. In the preface, Burton states that the second half of the book is wellsuited for student presentations and extra credit projects. I agree with this, but I would have liked a bibliography, or suggestions for further reading, at the end of each chapter (as in Gallian's Contemporary Abstract Algebra). This would be especially useful for anyone using the book for independent study or as a resource for a class presentation.
Overall I think this is a great book, appropriate for a number of different courses in number theory. His effort to provide the reader with the stories behind the mathematics is greatly appreciated.
John Cullinan is Visiting Assistant Professor of Mathematics at Colby College.
