Overview

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).

Full Product Details

Author:   Michael Th. Rassias
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   2011 ed.
Dimensions:   Width: 16.00cm , Height: 2.00cm , Length: 24.00cm
Weight:   1.450kg
ISBN:  

9781441904942

ISBN 10:   1441904948
Pages:   324
Publication Date:   02 December 2010
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

- 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.

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 MihAEilescu (EMS Newsletter March 2011)

From the reviews: Opening at random any page of this delightful book, the reader will almost certainly find something intriguing and interesting on the page. The book is an excellent training manual to use in preparation for (the number theoretic portions of) mathematical competitions and olympiads. The dedicated problem-solver would do well to avoid too early reference to the Solutions chapter. At the proper time, however, carefully studying this chapter is as rewarding as the earlier part of the book. This is a book that belongs in all academic libraries-from high school through graduate level. - F. J. Papp, Mathematical Reviews 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] 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 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 (of only 23 years), he is an outstanding specialist in this field. -Dorin Andrica, Zentralblatt MATH Containing all of the things he suggests one should know to compete successfully in an IMO competition. The book is based on his undergraduate thesis on computational number theory ... . it is so much more than basic things one should know, providing a rich overview of the many beautiful ideas in number theory. ... Plus, the text is enriched with historical comments, special problem-solving techniques, and a wealth of problems to investigate. ... Summing Up: Recommended. Academic readership, all levels. -Johnson, Choice, Vol. 49 (4), December, 2011

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 MihAEilescu, EMS Newsletter March 2011

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

Author Information

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.

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