##
**Number theory: course and solved exercises.
(Théorie des nombres: cours et exercices corrigés.)**
*(French)*
Zbl 0973.11002

Paris: Dunod. viii, 244 p. (1998).

As indicated in the Introduction, this book does not cover all the topics of Number Theory but deals mainly with Diophantine problems.

The list of the chapters is the following: Irrationality and Diophantine approximation, Expansion of real numbers and infinite products, Continued fractions, Quadratic number fields and Diophantine equations, Squares and sums of squares, Arithmetical functions, Padé approximants, Algebraic numbers and irrationality measures, Algebraic number fields, Ideals, Introduction to the methods of Transcendental number theory.

This list is much impressive when compared to the relatively modest size of the book, namely less than 250 pages. Moreover, each chapter ends with a list of a dozen of instructive exercises or more, and the detailed solutions of these exercises are given at the end of the book.

Some sections of this book are quite original and do not appear in standard introductions to Number Theory. For example: irrationality of the values of Tschakaloff’s function, Engel’s series for real numbers, infinite Cantor products, Padé approximants of \((1-x)^\alpha \), …and all these topics are quite interesting and fit very well in this book.

The aim of the author is always to give very precise and clear information. For example, he presents the continued fraction expansion of the number \(e\), Lambert’s expansion for \(\text{tg} (x)\), …The style is very clear. The arguments are very precise and almost always elementary. In my opinion, the level of presentation is convenient for graduate students, and this is an excellent book especially for future teachers in mathematics — the main target of the author. It gives the reader a beautiful concrete presentation of Number Theory and proposes to study more advanced books in this fascinating branch of mathematics.

The list of the chapters is the following: Irrationality and Diophantine approximation, Expansion of real numbers and infinite products, Continued fractions, Quadratic number fields and Diophantine equations, Squares and sums of squares, Arithmetical functions, Padé approximants, Algebraic numbers and irrationality measures, Algebraic number fields, Ideals, Introduction to the methods of Transcendental number theory.

This list is much impressive when compared to the relatively modest size of the book, namely less than 250 pages. Moreover, each chapter ends with a list of a dozen of instructive exercises or more, and the detailed solutions of these exercises are given at the end of the book.

Some sections of this book are quite original and do not appear in standard introductions to Number Theory. For example: irrationality of the values of Tschakaloff’s function, Engel’s series for real numbers, infinite Cantor products, Padé approximants of \((1-x)^\alpha \), …and all these topics are quite interesting and fit very well in this book.

The aim of the author is always to give very precise and clear information. For example, he presents the continued fraction expansion of the number \(e\), Lambert’s expansion for \(\text{tg} (x)\), …The style is very clear. The arguments are very precise and almost always elementary. In my opinion, the level of presentation is convenient for graduate students, and this is an excellent book especially for future teachers in mathematics — the main target of the author. It gives the reader a beautiful concrete presentation of Number Theory and proposes to study more advanced books in this fascinating branch of mathematics.

Reviewer: Maurice Mignotte (Strasbourg)

### MSC:

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

11Jxx | Diophantine approximation, transcendental number theory |

11Rxx | Algebraic number theory: global fields |

11Dxx | Diophantine equations |