*(English)*Zbl 1058.11001

This is the third edition of Guy’s well-known problem book on number theory (see Zbl 0474.10001 and Zbl 0805.11001 for previous editions), which is organised into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. Most of the problems are of an elementary, combinatorial nature. All of them can be understood without too much effort, but solving them is a different matter; indeed some of the problems are hopelessly difficult, including a few notorious ones. Being problems in number theory, even a complete solution to an original problem often generates yet more related problems. Thus many of the problems from the earlier editions have been expanded with more up to date comments and remarks. For most readers the bibliographies are the more useful aspects of the book; however, it has become so extensive for many of the problems that some pruning has become necessary. There are also several new problems, but the more interesting new feature is that, for about half of the problems, there is now a link of references to *N. J. A. Sloane’s* [Notices Am. Math. Soc. 50, No. 8, 912–915 (2003; Zbl 1044.11108)] very useful *Online Encyclopedia of Integer Sequences*.

As the author remarks, the book is perpetually out of date, but there are STOP PRESS entries. For example, in the problem concerning arithmetic progression of primes, reference is given for the recent announcement by Ben Green and Terence Tao that the primes contain arbitrarily long arithmetic progressions. There is little doubt that a new generation of talented young mathematicians will make very good use of this book, although it has to be said that the new edition can do with some copy-editing and proof-reading, because there are many easily detectable but irritating misprints.

##### MSC:

11-02 | Research monographs (number theory) |

00A07 | Problem books |

11Bxx | Sequences and sets of numbers |

11Dxx | Diophantine equations |

11Nxx | Multiplicative number theory |

11Pxx | Additive number theory; partitions |

11Axx | Elementary number theory |