This is a text for advanced undergraduates. It is notable for including *everything* that anyone would expect to find in an elementary number theory text, and more. For example, besides chapters on analytic number theory and elliptic curves, there is an outline of the resolution of Hilbert’s tenth problem. Continued fractions get a chapter, the gaussian integers are developed to answer the question of which integers are the sums of two squares, the Dirichlet convolution, of which I have always been fond, is given its due, and Hadamard matrices make an appearance.

Of course, this makes for a long book, more than 520 pages, that could not possibly be covered in a semester. As the authors point out, chapters 1–9, omitting chapter 4 on cryptography, give the basics, and then some, in 320 pages.

A feature of the book of which, for good or ill, prospective users should be aware is the inclusion of code, for both Mathematica and Maple, for various algorithms. Reading it did very little for me and , since the code can be found on the authors’ website, students would not have to retype it. Devotees of one or other of the programs may well feel differently.

The exposition is straightforward, written in good mathematical prose without distractions or infelicities. There is a good selection of exercises at the end of each of the five to ten sections in each chapter but no answers or hints at the back of the book, not even to problems whose numbers are congruent to 7 (mod 8). I would have liked to see more examples, but the authors are not aiming at weak students (who will find the book slow going, written as it is in lemma-proposition-theorem-proof style) and making the book longer would be a mistake.

The book is remarkably free of errors and misprints, for which the authors and their editor deserve great credit. Reviewers, however, are entitled to two free shots: there is a grammatical error on page 346, line 3, and on page 80 Lagrange is given a Spanish flavor (“Joseph-Luis”).

The book is pleasing to the eye except for the tables, which have too many heavy lines. The authors are to be congratulated for avoiding that uncouth usage, which unfortunately may be spreading, of “d*t*” in integrals.

It contains no color pictures, no sidebars, its margins are of the standard size, and though the authors include “real-life” applications, they do not claim that number theory will lead to lucrative careers.

Erickson and Vazzana have written an admirable text.

Woody Dudley’s number theory text was published forty years ago. The field has changed since then, slightly.