Mollin’s second edition takes “a truly elementary approach to number theory” that also entails “removal of all advanced material” so as “to be ... more accessible in scope.” As the mathematics remains rigorous, the value of the book to computing scientists, students, and practitioners remains very high, as I will elaborate below. I presume to suggest that the word “specialized,” instead of “advanced,” might have been used to describe the material that was removed from the first edition.

As I am a computing veteran, at least by virtue of age, and am currently teaching a follow-on to a formal methods course (on error detection and correction) that deals with the middle to end of a project’s life cycle, the substance and excellent pedagogy of this book are a serendipitous find for me. In the later parts of a life cycle, software is already 0s and 1s. Preservation of correctness throughout the iterative compile-link-load-execute phase of a project entails the application of well-chosen number-theoretic techniques, including several aspects of modular arithmetic; regarding the latter, chapter 2 of this book has one of the best treatments I’ve encountered.

The presentation of the integers modulo n (the congruence class) is first characterized by closure, commutativity, associativity, and identity-existence properties derived from chapter 1’s “Arithmetic of the Integers.” Mollin then introduces the terminology “commutative ring with identity” for any set having those properties. Such constructive sequencing characterizes the whole book and renders it very effective.

Speaking of sets, Appendix A, “Fundamental Facts,” treats this potential source of distractions in a virtuosic way by first defining well-definedness--“that it is always possible to determine whether or not a particular element belongs to a set”--and then defining a set to be “a well-defined collection of distinct objects.” This appendix then summarizes set theory and its operations and proceeds to other background topics: relations, functions, arithmetic and its axioms, inverses, ordering, some matrix theory, polynomial rings, vector spaces, and sequences and series.

The first two chapters comprise about one-third of the book and cover arithmetic and such fundamental topics as primes, tests for primality, and primes’ distribution; the (generalized) Chinese remainder theorem; partitions; generating functions; the ring Z/nZ and field Z/pZ; and divisors’ sums and numbers, including perfect numbers. This is as good a place as any to point out that there are, throughout the book, many well-placed and well-executed mini-biographies. For example, Biography 2.11 of Turing Award winner Michael Rabin tells of his work on a “mathematical foundation for finite automata theory,” within a sketch of Rabin’s career. Chapter 2 ends with cryptology and, by way of the Lester-Hill mini-biography, the exhortation and truism that a “rigorous mathematical approach [lies behind] today’s solid grounding of cryptography in mathematics.”

Chapter 3 defines and illustrates primitive roots with respect to two relatively prime integers. It follows smoothly from chapter 2, where the relevant Euler totient function was defined. The promise of application to (pseudo-)random number generation and to (public-key) cryptography is kept for later sections of this chapter. Chapter 4, on quadratic residues, also builds on its subject based on chapters 2 and 3’s introduction of quadratic congruences (and power residues). Mollin shows the usefulness of the Legendre symbol in the Lucas-Lehmer Mersenne primality test and in an elegant build-up to the quadratic reciprocity law. Chapter 4 concludes with a clear treatment of modern factoring algorithms.

The relevance to number theory of continued fractions is presaged in chapter 1, where they are mentioned in connection with Euclid’s algorithm. Chapter 5, on simple continued fractions, convergents, and Diophantine approximations, constructs a large context of surds and norm-forms around Pell’s equation(s) and concludes with factoring and various factoring methods’ computational complexity. Chapter 6 expounds on integers that are sums of powers, and includes Lagrange’s four-square theorem and Waring’s problem.

Diophantine equations are the subject of the final chapter 7, whose mere 20 pages impart significant expertise to the nonspecialist and a solid baseline and way forward to the prospective number theorist. Cohesion--a very good thing in software construction--is at work here, as chapters 1 to 6 are brought to bear. Norm-form equations are leveraged so as to support much of this chapter’s content. Appendices B and C both address computational complexity and primality testing, and are an excellent point of departure for algorithm designers.

Mollin’s second edition is an outstanding book among many excellent books on this most elegant and time-honored subject. It stands out along several axes: content, style, accuracy, and pedagogical efficiency. It would be difficult to make it anything other than a first choice.