This is a vast compendium of useful and interesting facts about prime numbers. On p. 428 the author writes, “In writing this book I wanted to produce a work of synthesis, to develop the theory of prime numbers as a discipline where the natural questions are systematically studied.” I think this is largely successful. The book is well organized and presents a thorough course in prime numbers. The publisher bills this as the third edition, although the title has changed: it is a revision of the 1989 second edition of *The Book of Prime Number Records*.

The “Records” in the title is a gimmick. Some of the facts are singled out as records, that is, the biggest, best possible, or best so far, according to some property, and there are occasionally long tables of primes with special properties, but these are a small part of the whole. Most of the book is a narrative about properties of primes and why they are important. The book generally quotes results without proof, but always gives a reference to the original paper, and often to later papers or books with simplified expositions. The bibliography is 75 pages (out of 550).

The book is a mixture of the expected and the unexpected. Among the expected, there is a lot about the distribution of prime numbers, twin primes, the Goldbach conjecture, and the Riemann zeta function. There are discussions of primes of special forms, such as Mersenne and Fermat primes and repunits. Among the less expected, there’s quite a lot about primality testing, including discussions of Carmichael numbers and of all kinds of pseudoprimes. There are discussions of primes with special properties, such as Sophie Germain primes. There’s a lengthy section on Lucas sequences (important for primality testing, but not often studied for themselves as they are here). The last chapter is more speculative and concerns heuristics, including a very nice summary of the Hardy and Littlewood conjectures and estimates in their Partitio Numerorum III.

Another book with a similar nature, but more discursive, is Shanks’s *Solved and Unsolved Problems in Number Theory*. Ribenboim’s *Little Book of Bigger Primes* is a selection of the items in the present book. It’s not that little, being about three-quarters the size of the present book, and it was revised more recently and is about eight years more up-to-date. Guy’s *Unsolved Problems in Number Theory* also gives a great deal of information about open problems in the primes.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.