Elementary Theory of Numbers (Dover Books on Advanced Mathematics)

Plus the sample problems and solutions in the above area.


Good Introduction to Key Topics and Proofs of Number Theory   April 21, 2006
Michael Wischmeyer (Houston, Texas)
14 out of 15 found this review helpful

William J. LeVeque's short book (120 pages), Elementary Theory of Numbers, is quite satisfactory as a self-tutorial text. It should appeal to math majors new to number theory as well as others that enjoy studying mathematics.

Chapter 1 introduces proofs by induction (in various forms), proofs by contradiction, and the radix representation of integers that often proves more useful than the familiar decimal system for theoretical purposes.

Chapter 2 derives the Euclidian algorithm, the cornerstone of multiplicative number theory, as well as the unique factorization theorem and the theorem of the least common multiple. Speaking from experience, I recommend that you take the time necessary to master Chapter 2, not just because these basic proofs are important, but more critically to reinforce the skills and discipline necessary for the subsequent chapters.

Two integers a and b are congruent for the modulus m when their difference a-b is divisible by the integer m. In chapter 3 this seemingly simple concept, introduced by Gauss, leads to topics like residue classes and arithmetic (mod m), linear congruences, polynomial congruences, and quadratic congruences with prime modulus. The short chapter 4 was devoted to the powers of an integer, modulo m.

Continued fractions, the subject of chapter 5, was not unfamiliar and yet, as with congruences, I quickly found myself enmeshed in complexity, wrestling with basic identities, the continued fraction expansion of a rational number, the expansion of an irrational number, the expansion of quadratic identities, and approximation theorems.

I have yet to tackle the last two chapters, the Gaussian integers and Diophantine equations, but my expectation is that both topics will also require substantial effort and time. LeVeque's Elementary Theory of Numbers is not an elementary text, nor a basic introduction to number theory. Nonetheless, it is not out of reach of non-mathematics majors, but it will require a degree of dedication and persistence.

For a reader new to number theory, LeVeque may be too much too soon. I suggest first reading Excursions in Number Theory by C. Stanley Ogilvy and John T. Anderson, another Dover reprint. It is quite good.

Some caution: LeVeque emphasizes that many theorems are easy to understand, and yet this very simplicity is a two-edged sword. Simple theorems often provide no clues, no hints, on how to proceed. Discovering a short and elegant proof is often far from easy. LeVeque also stresses that a technique ceases to be a trick and becomes a method only when it has been encountered enough times to seem natural. A reader new to number theory may initially be overwhelmed by the variety of techniques used.

A nit: The Dover edition of LeVeque's Elementary Theory of Numbers would benefit from a larger font size. I occasionally found myself squinting to read tiny subscripts and superscripts.


Readable, clear, but needs an errata page   September 3, 2007
Kevin O'Gorman (California, USA)
As others have said, this is a fairly easy read. For me it's actually fun and I'm working through it for that reason. But:

- I don't normally use a highlighter, but found it necessary to highlight symbols where they were defined, because some of them come up only once in a while and it's easy to forget where the definition is. Symbols are not indexed. I have started my own symbol index in the back of the book.

- There are some annoying errors. The theorem to be proven in section 1-3, problem 2 is false for n=1. The decimal expansions in the chapter on continued fractions (page 75) are wrong (for example 1.273820... should actually be 1.273239...). It seems to me if you're going to give 7 digits they should be the right 7 digits.

On the other hand, these errors don't affect the overall flow of the text, and I'm having a great time working through this book on my own. I've read through the whole thing over the summer, and I'm going back through doing problems and writing programs. I was a math major 40 years ago, and haven't done much with it since, to give a context for that remark.