In the preface to this book, the author asserts (p. xii) that:

Many other mathematical subjects, calculus, for example, would have undoubtedly evolved much as they are today quite independent of the individual people involved in the actual development, but number theory has had a wonderfully quirky evolution that depended heavily on the particular interests of the people who developed the subject over the years.

This forms the organizing principle for this fine survey of elementary number theory through the perspective of the people who were involved in the development of its fundamental concepts. Thinking about this choice, I’m not sure that the end result — number theory as a branch of mathematics as it stands today — would be meaningfully different if different people had contributed to the subject. We might not talk about “Mersenne primes”, for example, but it seems a reasonable premise that prime numbers of the form 2^{n} – 1 would have attracted interest somewhere along the historical timeline. (I do agree with his point about calculus.)

The net result of this choice, though, is an excellent contribution to the list of elementary number theory textbooks. Number theory, it is true, has as rich a history as any branch of mathematics, and Watkins has done terrific work in integrating the stories of the people behind this subject with the traditional topics of elementary number theory. There is more than enough material here for a one-semester course, and while this is standard for textbooks at this level, the added historical and biographical material — which cover mathematical developments and people well into the 20^{th} century — are well worth the increased weight of the text.

Mark Bollman (mbollman@albion.edu) is professor of mathematics and chair of the department of mathematics and computer science at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. Mark’s claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.