Number Theory: Concepts and Problems

XYZ Press is a small publishing company whose books are distributed by the American Mathematical Society, and which specializes in texts that are designed to teach mathematical problem solving and competition preparation. It was established by Titu Andreescu, who is also the editor, author or co-author of many of the roughly two dozen texts published by this company, including the one now under review, which deals with problems in number theory.

I have seen a good many problem books over the years, and must admit that after a while many of them tend to blur together in my mind. This one, however, is something of an exception, and stands out more vividly from the crowd, both because of its heft (at almost 700 pages, it is quite thick) and, more significantly, because of its content: in addition to setting out problems and solutions, it actually develops a substantial amount of elementary undergraduate number theory, and therefore seems more textbook-oriented than many other problem-style books. As a result, it is, I think, more versatile, with a larger target audience, than is typical of such books.

Chapter 1 of the text is just a two-page preface, summarizing the origins and contents of the text. Chapters 2 through 7, which comprise roughly two thirds of the text, deal with substantive number theory, covering most of the “usual suspects” of a standard undergraduate course (divisibility, primes, the Fundamental theorem of arithmetic, congruences, quadratic residues and nonresidues, some Diophantine equations, and arithmetic functions), along with a number of topics that are not typically covered in such a course (for example: Bertrand’s postulate and the distribution of primes, a brief discussion of the abc conjecture, and p-adic valuations).

As noted above, each of these six chapters focuses both on the development of the underlying theory and on problems. Basic definitions are provided, theorems are stated precisely, and proofs are given, including of such results as the law of quadratic reciprocity. In addition to this, however, many solved problems (“examples”) appear in the text, and in addition each of these chapters ends with a section consisting of a substantial number, typically 50 or more per chapter, of “practice problems”. These practice problems often seem to have a somewhat different flavor than standard textbook homework problems (they are more in the nature of competition problems than homework problems, and hence more difficult), but there is some occasional overlap. Solutions to these practice problems constitute chapter 8 of the text, which is more than 200 pages long.

Because of the format of the book, and specifically because number theory is developed pretty much from scratch rather than just assumed, this book should be of interest to people who are taking or teaching a basic undergraduate number theory course, even if they have no interest in ever entering a mathematics Olympiad-style competition. Certainly any faculty member teaching such a course would likely find this book a treasure trove of interesting and challenging problems for use as homework or (if the students are unlucky!) exam questions. (Since a number of the problems have a combinatorial flavor, people taking or teaching a course in combinatorics might also find something useful here.)

The book could also potentially serve as the main text for a course in number theory, although in some respects this might be problematic. For one thing, some topics that are fairly standard fare in a number theory course are not covered here. Continued fractions are not discussed, and there is also no discussion of applications of number theory to cryptography, a topic that I think students in a basic number theory course really enjoy seeing. In addition, the last time I taught number theory I also spent a few weeks on quadratic rings where unique factorization fails, both to illustrate the importance and significance of the fundamental theorem of arithmetic, and to give the students at least a glimpse of algebraic numbers. This book doesn’t include such material, either. (Books that do discuss this material include Stillwell’s Elements of Number Theory and An Introduction to Number Theory and Cryptography by Kraft and Washington.)

The most negative feature of this book, one that also militates against its use as an actual text for a number theory course, is the total lack of an index. It is incomprehensible to me how a book intended for college students can be published without one, but this seems to be fairly common for books published by XYZ Press: although I have not reviewed the contents of every book published by XYZ, I did glance at the Table of Contents for about half a dozen chosen at random, and none of them indicated the presence of an Index.

But let us not end this review on a sour note. This book is a valuable collection of problems in elementary number theory; if I ever teach the course again, I will keep it close at hand.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.