The study of properties of prime numbers is one of the oldest topics in mathematics. Here it is easy to ask questions that are hard, often hopeless, to answer. You will find a large collection of such questions in this book. One has to use sophisticated methods to prove the deep properties of primes, but primes are also suitable for computer experiments. Last but not least, primes are used in cryptology.

Four years ago, Ribenboim published a book about prime numbers[1]. This volume is its little brother: shorter, without technical details and long hard proofs, but including all important aspects. It is up to date, with many references. A minor criticism is that the author hints at his earlier  work [1]  in the preface, but I was not able to find its dates in the references.

As mentioned, this book includes all important aspects of prime number theory: classical and modern proofs for the infinitude of the number of primes; various primality tests and factorization methods; prime-based cryptographic schemes; prime defining functions; analytical investigations concerning the primes, such as counting and related functions; and special kinds of primes. The author presents this rich material within a few pages in such a way that the presentation is always fluent but rigorous. The definitions and theorems are precisely formulated, but most of the theorems are stated without proof. In return for this, you find up-to-date examples and records.

Computer experiments are becoming more important in mathematics and prime number theory. The examples and records in Ribenboim’s book are the results of long experiments. Ribenboim not only presents this rich material, but also gives its theoretical interpretation.

This book is well documented. Original papers and books are correctly cited. The list of references is 26 pages long. I find the ordering of the references--chapters, year of publication, then name of the first author--complicated. It is easier for me to find a reference under the name of the first author.

Ribenboim did conscientious work in collecting the most important results about prime numbers. Nevertheless, specialists would prefer to see other theorems in this book. I shall mention only two. On page 133, the Mertens function is introduced, but nothing is said about Mertens’s famous conjecture and its refutation by Odlyzko and te Riele [2]. On page 172 the author writes, “It is not known if such (different from one) repunits can be a k-th power, for any k not a multiple of 2, 3 or 5.” He should have cited Shorey and  Tijdeman’s  proof that only finitely many effectively computable repunits that are perfect powers  exist [3]. 

I enjoyed Ribenboim’s book and can recommend it to all those who are working in or studying number theory or its applications.