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Number Theory: Structures, Examples, and Problems

Titu Andreescu and Dorin Andrica
Publication Date: 
Number of Pages: 
[Reviewed by
Mehdi Hassani
, on

Elementary ideas about whole numbers, such as divisibility, prime numbers and congruences, are one of the main subjects for questions in various exams, especially in Mathematical Olympiads. Interested students preparing for such exams often cannot find satisfying problems in usual text books, or even in most problem books. It seems that the aim of this book is to solve this problem.

The book is a collection of number theory problems chosen from various national and international Mathematical Olympiads. T. Andreescu and D. Andrica, authors of the book, are both veterans of the International Olympiad; they have written other books in other areas for Olympiad exams. I believe that the expert authors know exactly what is important for number theory problems in Olympiads, and what they should discuss for interested students.

Each chapter of this book starts a brief and fast review of concepts, with various solved examples, mainly selected from Olympiad exams. Then come many other similar problems as exercises, with complete solutions in the second part of the book.

The book could be used as a text for undergraduates, but the teacher would have to add easier examples in order not to overwhelm most of the students. The nain audience will consist of Olympiad-level students, of course. Some of problems, however, have the potential to become the subject of undergraduate research. I recommend this friendly volume for students looking for challenging problems in number theory and teachers of number theory for undergraduates, who can use it as a source of highly tricky examples.

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.


Preface.- Divisibility.- Powers of Integers.- Floor Function and Fractional Part.- Digits of Numbers.- Basic Principles in Number Theory.- Arithmetic Functions.- More on Divisibility.- Diophantine Equations.- Some Special Problems in Number Theory.- Problems Involving Binomial Coefficients.- Miscellaneous.- Glossary.- References.- Index.