Problem-Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads

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Michael Th. Rassias

Problem-Solving and Selected Topics in Number Theory

In the Spirit of the Mathematical Olympiads

Michael Th. Rassias

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English

Springer-Verlag New York Inc.
02 December 2010

Number theory

The book provides a self-contained introduction to classical Number Theory. All the proofs of the individual theorems and the solutions of the exercises are being presented step by step. Some historical remarks are also presented. The book will be directed to advanced undergraduate, beginning graduate students as well as to students who prepare for mathematical competitions (ex. Mathematical Olympiads and Putnam Mathematical competition).

By: Michael Th. Rassias
Imprint: Springer-Verlag New York Inc.
Country of Publication: United States
Edition: 2011 ed.
Dimensions: Height: 240mm, Width: 160mm, Spine: 20mm
Weight: 1.450kg
ISBN: 9781441904942
ISBN 10: 1441904948
Pages: 324
Publication Date: 02 December 2010
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

- Introduction.- The Fundamental Theorem of Arithmetic.- Arithmetic functions.- Perfect numbers, Fermat numbers.- Basic theory of congruences.- Quadratic residues and the Law of Quadratic Reciprocity.- The functions p(x) and li(x).- The Riemann zeta function.- Dirichlet series.- Partitions of integers.- Generating functions.- Solved exercises and problems.- The harmonic series of prime numbers.- Lagrange four-square theorem.- Bertrand postulate.- An inequality for the function p(n).- An elementary proof of the Prime Number Theorem.- Historical remarks on Fermat’s Last Theorem.- Author index.- Subject index.- Bibliography and Cited References.

Michael Th. Rassias has received several awards in mathematical problem solving competitions including two gold medals at the Pan-Hellenic Mathematical Competitions of 2002 and 2003 held in Athens, a silver medal at the Balkan Mathematical Olympiad of 2002 held in Targu Mures, Romania and a silver medal at the 44th International Mathematical Olympiad of 2003 held in Tokyo, Japan.

Reviews for Problem-Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads

Number Theory problems are among the most tricky in Mathematical Olympiads (MO). For students who are going to participate in such a tournament, and also for their teachers, a book that covers the main topics of fundamental number theory and contains various problems related to MOs is very useful. The book under review is exactly such a friendly volume, arranged in two main parts: Topics and Problems...The book under review is not the only book which focuses on olympiad problems in number theory, but because of its structure (containing topics and problems), it is also useful for teaching. I highly recommend this book for students and teachers of MOs. --Mehdi Hassani, MAA Reviews This book has a few particular characteristics which make it unique among similar problem books... [He] focused on problems of number theory, which was the field of mathematics that began to capture his passion. It appears like a confession of a young mathematician to students of his age, revealing to them some of his preferred topics in number theory based on solutions of particular problems... Michael does not limit himself to those particular problems. He also deals with topics in classical number theory and provided extensive proofs of the results...It offers pleasant reading for young people who are interested in mathematics. They will be guided to easy comprehension of some of the jewels of number theory. --Preda Mihailescu, EMS Newsletter March 2011 Number theory is one of the most active and important fields in Mathematics with a substantial and large variety of applications in several disciplines including representation theory, cryptography, coding theory, dynamical systems, [and] theoretical physics. The present book provides a wonderful presentation of concepts and ideas as well as problems with their solutions in Number Theory. Although most of the problems solved in this book were given in international mathematical contests and hence are of high level of complexity, the author has succeeded in providing solutions and extensive step-by-step proofs in a rigorous yet very simple and fascinating way. Even though the author is a very young mathematician! he is an outstanding specialist in this field. --Dorin Andrica, Zentralblatt MATH