If you enjoy reading popular mathematics books, you will probably enjoy Calvin C. Clawson's Mathematical Mysteries. The author has an engaging style of writing, and his enthusiasm for the subject shines through. Topics discussed include prime numbers (of course), number sequences (eg. The Fibonacci sequence), the golden ratio, the proof that square root of 2 is irrational, primes and secret codes, Ramanujan, Goldbach's Conjecture, the Riemann Zeta function, and Godel's incompleteness theorem. There's a good bit of history included throughout, and there is even a chapter that discusses "Numbers and the Occult." The topics do become weightier by the end of the book, and at that point it may be only mathematicians who are still reading. On the whole, most readers of MAA Online will find most of the topics familiar.
I enjoyed the book on a couple of different levels. I am teaching a liberal arts mathematics course this semester, and Clawson's book treats many of the topics we are discussing in that class, from a slightly different perspective. As a community college professor, his communication skills are well honed. I assigned his chapter on secret codes to a student who is doing a research project. It is the most readable account of RSA codes I have seen. From a mathematician's perspective, I also enjoyed his treatment of the Riemann Zeta function and his sketch of the proof of Godel's theorem. Not being a number theorist or a logician myself, he included enough details to remind me why these topics are interesting.
Of course the intended audience is not professional mathematicians. Clawson does not prove every result he discusses (or even hint at a proof in many cases). In the Ramanujan chapters in particular, he only justifies a small fraction of the equations he presents. This I found a little bit unsatisfying. Clearly, he does this to keep non-mathematicians from getting frustrated. With this caveat in mind, those readers interested in more details should know how to find them.
In summary, I found Clawson's book to be a very enjoyable read, a good presentation of familiar and some less familiar topics. This book would serve as an excellent resource for one who is teaching an elementary course that deals with number theory.
Joel Foisy (firstname.lastname@example.org) is an assistant professor of mathematics at SUNY Potsdam in Potsdam, NY.