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Elements of the theory of numbers. (English) Zbl 0932.11001
San Diego, CA: Harcourt/Academic Press. xviii, 517 p. \$ 39.95 (1999).

In most institutions of higher learning, number theory is not a required subject for math majors. As a consequence, the total market for number theory textbooks is only about 5000 copies per year in the entire United States and Canada. Nevertheless, the proliferation of number theory textbooks continues unabated.

The most striking feature of this particular text is the authors’ lavish use of abstract algebra. Indeed, the book contains a mini-course in algebra: Sections 2.3, 3.3, and 3.4 are devoted to groups, rings, fields respectively. A well-known theorem in number theory states that if $a$ and $b$ are natural numbers, then there are integers $x$, $y$ such that $\left(a,b\right)=ax+by$. On page 62, the authors present a slightly weaker version of this theorem, with a proof based on group theory. In the opinion of this reviewer, an experienced teacher of elementary number theory, this algebra-laden approach to number theory is more likely to bring confusion than enlightenment to the mind of the student, unless the instructor is blessed with students of exceptional ability.

The authors seem to be fond of off-beat proofs. In Chapter 1, they present an interesting proof of the divergence of the sum of the reciprocals of the primes. In Chapter 2, they offer a complicated proof of the fundamental theorem of arithmetic that is based on well-ordering rather than on Euclid’s Lemma. In Chapter 7, the authors’ proof of Euler’s theorem regarding the form of even perfect numbers is excessively long.

Chapter 6, which is entitled “Residues”, deals not only with quadratic, but with cubic and quartic residues. The latter topics require the introduction of some algebraic number theory. Like most other number theory texts, this one fails to present an algorithm for solving quadratic congruences. The chapter on congruences contains the now customary homage to cryptography.

The first seven chapters of this lengthy book (518 pages) are called “Fundamentals”. The final four chapters are devoted to “Special Topics”, namely sums of squares and other representational problems, number fields, partitions, and recurrences. The authors deserve applause for their inclusion of a chapter on partitions, an undeservedly neglected topic in number theory. This text contains numerous exercises, but some of them are trivial, e.g. (1) Show that the function $x/lnx$ is increasing for $x>c$; (2) Corroborate the stated number of digits for ${M}_{2976221}$; (3) Show that a Mersenne prime is a $\left(4k+3\right)$ prime.

In summary, learning from this book might be a worthwhile experience for students of above-average skill and ambition. It is most likely overly sophisticated for students with more modest goals and backgrounds.

##### MSC:
 11-01 Textbooks (number theory) 11Axx Elementary number theory 11P81 Elementary theory of partitions 11B37 Recurrences 11R11 Quadratic extensions