I really like this book. I may never have the chance to use it as a textbook, but it sits on a nearby shelf every time I teach number theory. I use it as a source for ideas, examples, and problems. And sometimes, when I’m unhappy with the proof in our textbook, I will check to see if Flath has found the “right” proof. He often has.

As Flath explains in the introduction, the book was born when he was asked to teach a number theory course for senior undergraduates in Singapore. This allowed him to assume much more mathematical maturity than in the typical American undergraduate course in number theory, and he decided to teach a course inspired by Gauss’s *Disquitiones Arithmeticae* and especially the theory of binary quadratic forms developed there.

The book is an introduction, starting with the usual material: primes, unique factorization, linear diophantine equations, integers modulo *m*. But the pace is very fast: by page 34 we are talking of Gaussian integers and Minkowski’s theorem (in two dimensions) is on page 48. Quadratic reciprocity is proved on page 77. The proof given is Gauss’s first; the proof using Gauss’s Lemma that typically appears in undergraduate textbooks (but done better than usually) follows in the next section. By the end of the book we are studying the group of classes of binary quadratic forms and genus theory.

The language of abstract algebra is used throughout, as are series expansions and elementary real analysis. Flath is not afraid of difficult proofs. There are many good problems, some of them quite difficult. (And there is no instructor’s manual!)

The writing is terse but clear. I have often found that Flath’s account of some topic is easier to understand than others. Most of all, Flath has good mathematical taste: the material he covers is beautiful and definitely worth learning.

In American universities, elementary number theory tends to be taught to students earlier in their mathematical career, using mostly elementary methods. This is not wrong: it exposes students to some beautiful mathematics that is nevertheless within their reach. But there’s a downside: avoiding algebra and analysis means that one must often replace a more structural proof by a more elementary one based on some ingenious trick. It’s easy to leave the impression that number theory boils down to finding such tricks.

Flath’s book offers an alternative: using the basics of analysis and algebra to give a somewhat deeper account of (still) elementary number theory. With some judicious skipping of the material in the first few pages, it would make an excellent capstone course for mathematics majors or a great introduction to number theory for master’s students. I’m very happy that AMS/Chelsea decided to reprint it.

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Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He is currently the editor of MAA Reviews but is happy that his term is ending soon.