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Elements of the Theory of Numbers

Thomas P. Dence and Joseph B. Dence
Academic Press
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
, on
This book, published in 1999 but still in print, is a text for a fairly high-end course in elementary number theory at the junior/senior level. The authors have an interesting point of view, but, both in the choice of topics covered and manner of presentation, this book is somewhat more demanding than most contemporary competing texts. 
The book is divided into two parts, the first of which, “The Fundamentals”, consists of seven chapters. The first two of these discuss divisibility, prime factorization, and the greatest common divisor. The next four involve aspects of modular arithmetic (congruences, polynomial congruences, primitive roots, and residues). The final chapter is on arithmetic functions and related issues such as perfect numbers.
All of the standard material on these topics is covered here, and there is also some coverage of material that is not so standard at this level. For example, the chapter on residues discusses not only quadratic reciprocity but cubic and quartic reciprocity as well. Also, concepts from abstract algebra (groups, rings, fields) are introduced fairly early (groups are defined on page 48), discussed in detail and used at length when discussing such things as the greatest common divisor, Fermat’s little theorem, and Euler’s theorem. Some analytic results about the prime numbers (for example, the theorem that the sum of the reciprocals of the primes diverges) are proved fairly early. Even basic results are sometimes proved in nonstandard ways: most books at this level, for example, deduce the Fundamental Theorem of Arithmetic from Euclid’s Lemma, but here the order of presentation is reversed. 
The second part of the book (“Special Topics”) contains four chapters, one on “representation problems” (Pythagorean triples, sums of two and four squares, and other similar problems), the second an introduction to algebraic number fields, and the final two on partitions and recurrence relations, respectively. I associate these topics more with combinatorics than number theory, but their inclusion here certainly offers some interesting options for an instructor.
As in the first part of the book, algebra is used extensively, particularly (of course) in the chapter on algebraic number fields. This chapter goes beyond ad hoc discussion of particular rings of algebraic integers (such as the Gaussian integers) but instead develops a lot of introductory field theory, including, for example, a proof of the Primitive Element Theorem. And, again as in the first part of the book, there are some unusual proofs here; the four-square theorem is proved using matrices (including the Hermitian adjoint) with Gaussian integer entries. 
I did notice some omissions. Fermat’s Last Theorem, for example, is given very little discussion; it is mentioned only in a couple of sentences. Likewise, sums of three squares are not discussed at all; while I completely agree with the authors’ decision not to prove the theorem that characterizes them (it is complicated and not a lot of fun), I also think that in making the jump from two squares to four squares it would have been nice to at least mention three squares and state (without proof) the relevant theorem. Continued fractions are not covered in the kind of depth that they are in some other texts. Likewise, the subject of cryptography, which is nowadays often viewed as a major part of any undergraduate number theory course, is dealt with here in about eight pages of text. Another topic that is occasionally mentioned in some current number theory texts is an introduction to elliptic curves, but these are not mentioned here. 
Nevertheless, there is much to like about this book, which holds up quite well despite the fact that it necessarily omits some recent results in number theory (such as the proof of the Bounded Gaps Conjecture). The writing is clear and often elegant, and the unusual presentation makes for an interesting account. There are lots of examples illustrating the theorems, and also lots of exercises. The exercises cover a wide range of difficulty level: some are extremely simple; others are more challenging.  Each chapter ends with a good bibliography, covering articles as well as texts, and all of which have the added advantage of being annotated. 
Notwithstanding these good features, however, I would not select this book as a text for the undergraduate number theory course at Iowa State University. It is no secret, I think, that undergraduates today are generally less prepared for college mathematics classes than were their counterparts of twenty years ago, so books that might have made a good textbook back then may well be too demanding today for average-ability students. That is particularly true here, given the authors’ extensive use of abstract algebra (which is not a prerequisite for the number theory course at Iowa State, or at many other universities).  However, the book’s interesting features and nice exposition fully justify its Basic Library List double-star rating. 


Mark Hunacek ( teaches mathematics at Iowa State University. 


Part I The Fundamentals
Introduction: The Primes
The Fundamental Theorem of Arithmetic and Its Consequences
An Introduction to Congruences
Polynomial Congruences
Primitive Roots
Multiplicative Functions
Part II Special Topics
Representation Problems
An Introduction to Number Fields
Recurrence Relations

Appendix I: Notation
Appendix II: Mathematical Tables
Appendix III: Sample Final Examinations
Appendix IV: Hints and Answers to selected problems