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.