Read This! The MAA Online book review column Discovering Number Theory by Jeff Holt and John Jones Reviewed by Donald L. Vestal

Discovering Number Theory is a textbook on number theory and, as the title suggests, the emphasis is on the discovery aspect. The book covers the usual topics in an undergraduate number theory course, but does so with a great deal of assistance from the computer. And the focus, unlike that of many current mathematical texts, is on letting the students do much of the exploration themselves. A passage from the Introduction to the Instructor's Edition describes it as follows:

As one of our undergraduate professors was fond of saying [of math texts], "What you see here is the final product. However, you should be aware that there was a huge pile of wadded up paper tossed in the corner that contained the work leading up to what you see."

One goal we have for this text is to allow students to see and create their own pile of wadded up paper.

The arrangement of this book allows for just that.

Each chapter consists of four parts: the prelab, the lab, the summary, and the homework section. The student begins with the prelab to get them thinking about a particular topic. Some definitions are given, along with a few problems or examples, and then the students are directed to the lab. The student has a choice of three different ways to do the lab, depending on what kind of computer software they have access to: Maple notebooks, Mathematica notebooks, or HTML/Java Applets. These are provided on a compact disc that comes with the book. The lab guides the student through the main objective of the chapter. The chapter topics are

1. Divisibility and Factorization
2. The Euclidean Algorithm and Linear Diophantine Equations
3. Congruences
4. Applications of Congruences
5. Solving Linear Congruences
6. Primes of Special Forms
7. The Chinese Remainder Theorem
8. Multiplicative Orders
9. The Euler phi-function
10. Primitive Roots
11. Quadratic Congruences
12. Representation Problems
13. Continued Fractions

Contained in each lab are several research questions for the student. Some ask for conjectures, some ask for proofs. After the lab is completed, the instructor has the option of having the students write up their results in a lab report. Once the lab work is finished, the instructor gives the student the chapter summary (not included with the student version of the text). The summary gives the answers/proofs to the Research Questions, along with additional explanations for the student who is interested in "Going Farther." The last section of the chapter is the homework section. Lots of problems are given, some computational, some theoretical.

The topics are well presented. The programs in the labs are well written, allowing students to plug in many numbers to try to construct and test conjectures. Indeed, students are encouraged to create and prove conjectures based on the patterns they find in the labs. So the book is well-designed for and upper level undergraduate course.

Since the labs are presented in three different formats, many of the pages in the book are going to be redundant and useless for the student. To compensate, the pages are perforated and hole-punched, so that the student can read and make notes on the pages, rip them out, and put them in a binder for study. The excess pages can be torn out and discarded, or given to someone else who may be using a different format.

Each book comes with a compact disc with the Maple, Mathematica, and HTML/Java versions of the various labs. The instructor's version comes with the instructor's CD, which has all of the labs, an instructor's guide to the book, the chapter summaries, and the solutions to the odd-numbered homework problems. The guide and solutions are available in both PostScript (ps) and portable document file (pdf) form, while the chapter summaries are available in dvi, ps, and pdf format. One note of caution: the student version of the book does not contain the chapter summaries. So if you use this as a textbook, you may have some panicking students on the first day of class complaining, for example, about how their book is missing pages 85 through 90.

Some specific high/low points: The rock game in chapter 4 is a cute way to introduce congruences. A fair amount of time is spent on cryptography (a plus as far as I'm concerned, but others may disagree). As an added bonus, one of the lab sections on cryptography has something that more math texts could use: a P. J. O'Rourke quote. On the down side, a couple of the Mathematica programs didn't seem to work as advertised (at least not on Mathematica version 4.0): specifically, the sum of squares program (sumofmany) from the chapter 12 lab and the Pell's Equation solver (guesssolns) from the chapter 13 lab. The Java versions seem okay, though. (I didn't do the Maple versions, so I can't make comments on those.)

Publication Data: Discovering Number Theory, by Jeff Holt and John Jones. W. H. Freeman & Company, 2001. Paperback, 649 pp, $80.55. ISBN 0-7167-4284-5.

Donald L. Vestal is an Assistant Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, reading, and listening to the music of Rush. He can be reached at vestal@mwsc.edu.

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