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**Number theory: concepts and problems.**
*(English)*
Zbl 1445.11001

Allen, TX: XYZ Press (ISBN 978-0-9885622-0-2/hbk). viii, 686 p. (2017).

The book under review constitutes a masterful introduction to number theory. The content is very nicely presented as well as treated with a plethora of examples and problems which will be of interest not only to uninitiated readers but also to readers who already have a background in number theory. Each example and problem is very carefully chosen to present a nice technique or method which assists the reader to comprehend the theory but also to obtain a strong skill set very much enhancing his/hers problem-solving skills.

The present book will also be very useful to students preparing for Mathematical Olympiads as it includes a broad variety of problems of Olympiad caliber. As mentioned above, the problems are presented in such a manner that the reader understands the underlying ideas and learns methods and techniques which will be invaluable for solving Olympiad type problems in number theory. Specifically, the chapters are devoted to: divisibility, Bezout’s theorem, Gauss’ lemma, Diophantine equations, the fundamental theorem of arithmetic, congruences involving prime numbers, \(p\)-adic valuations and the distribution of primes, as well as congruences of composite moduli, presenting a wealth of related topics in each chapter.

The choice of topics covered is fairly rich, going through most of the beautiful gems of number theory.

Overall, I believe that this book belongs to all libraries of universities as well as to the book collection of all mathematics enthusiasts.

The present book will also be very useful to students preparing for Mathematical Olympiads as it includes a broad variety of problems of Olympiad caliber. As mentioned above, the problems are presented in such a manner that the reader understands the underlying ideas and learns methods and techniques which will be invaluable for solving Olympiad type problems in number theory. Specifically, the chapters are devoted to: divisibility, Bezout’s theorem, Gauss’ lemma, Diophantine equations, the fundamental theorem of arithmetic, congruences involving prime numbers, \(p\)-adic valuations and the distribution of primes, as well as congruences of composite moduli, presenting a wealth of related topics in each chapter.

The choice of topics covered is fairly rich, going through most of the beautiful gems of number theory.

Overall, I believe that this book belongs to all libraries of universities as well as to the book collection of all mathematics enthusiasts.

Reviewer: Michael Th. Rassias (Zürich)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11Axx | Elementary number theory |

11N05 | Distribution of primes |

00A07 | Problem books |