Problem-Solving and Selected Topics in Number Theory

This self-contained introduction to classical number theory is written for undergraduate and graduate students, alike. Proofs of the theorems and solutions of the exercises are presented step-by-step.

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).

Presents the historical background of various topics in number theory; Provides a self-contained introduction to classical number theory; Includes step-by-step proofs of theorems and solutions to exercises; Designed for undergraduate students, particularly those who would like to prepare for mathematical competitions. Includes supplementary material: sn.pub/extras

Autorentext
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.

Klappentext
This book is designed to introduce some of the most important theorems and results from number theory while testing the reader's understanding through carefully selected Olympiad-caliber problems. These problems and their solutions provide the reader with an opportunity to sharpen their skills and to apply the theory. This framework guides the reader to an easy comprehension of some of the jewels of number theory The book is self-contained and rigorously presented. Various aspects will be of interest to graduate and undergraduate students in number theory, advanced high school students and the teachers who train them for mathematics competitions, as well as to scholars who will enjoy learning more about number theory. 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.

Zusammenfassung

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).

Inhalt

• 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.