Customer Reviews

Discovering Number Theory w/CD-ROM

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I have used these materials to teach an undergraduate course in number theory,
and I think they are an excellent way to give independent and motivated
students an experience with research. This book is just for the students
to use, and so it is deliberately incomplete. Instructors should contact
the publisher to receive the supplements.

This book is NOT designed to be a reference text.

The basic idea is the following: Let students discover and prove the main
results, then extend the material to include topics they couldn't reasonably
do by themselves.

A typical chapter includes a prelab, lab, chapter summary (instructor supplement),
and problems. The prelab introduces the basic ideas, and then in the lab,
students conjecture and prove the main results of the chapter. When they
are finished, they receive a chapter summary (available in the instructor
supplements), including lab results. Obviously it would be foolish to include
lab answers in the student material. Each chapter also includes homework problems.

The instructor materials also include Going Father sections, which include
material that would be too difficult for students to develop on their own
(e.g., Rabin-Miller). Taken as a whole, the entire package, student book
and instructor supplements, is comparable to the typical undergraduate textbook.

The CD includes the labs (for Mathematica, Maple, and the web), so there wasn't
any practical need to print them in the book, in triplicate. Perhaps they could
have included the Going Farther sections instead. The prelabs are fine as
written, but I supplement a few. The homework is minimal, but adequate, as
students have a lot of work to do in the labs.

In the preface, the authors state that this book is designed to be used in a one-semester upper division undergraduate course in number theory. I would dispute that claim, in my opinion the level of mathematics is not high enough for an advanced undergraduate course. There are not enough examples where the reader is required to work their way through a formal proof.
The chapters are:

*) Divisibility and factorization
*) The Euclidean algorithm and linear Diophantine equations
*) Congruences
*) Applications of congruences
*) Solving linear congruences
*) Primes of special forms
*) The Chinese remainder theorem
*) Multiplicative orders
*) The Euler phi-function
*) Primitive roots
*) Quadratic congruences
*) Representation problems
*) Continued fractions

which certainly gives the impression that there is depth to the coverage. However, that is not the case. Each chapter begins with a few pages of what is called prelab. There are a few statements about the topic and then the student is to proceed to working a set of lab exercises.
For each chapter there is a set of lab exercises in triplicate, they are identical, except one is in Maple, one in Mathematica and the last runs Java applets in a browser. This repetition is a bit puzzling, as everyone has access to a browser. There is a set of homework problems at the end of each chapter with no solutions included. With the labs in triplicate and there being only a few pages of explanation of the mathematical background for those labs, there is nowhere near the depth of coverage that I would expect in an advanced undergraduate class.
However, I will say that this book would be an excellent text for any supplemental lab that you might want to do with a number theory course.