Problem-Solving and Selected Topics in Number Theory
In the Spirit of the Mathematical Olympiads
Rassias, Michael Th.
2011, XIV, 324 p.
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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.
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.
Content Level »Lower undergraduate
Keywords »Mathematics Competition - Mathematics Olympiad - Number Theory - Problem-Solving
- 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.- Bibliography and Cited References.- Author index.- Subject index.