Problem-Solving and Selected Topics in 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.

• - 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.

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