Instructor: Dr. Csaba SZABÓ

Text:  Selected chapters of Melvyn B. Nathanson: Elementary Methods in Number Theory and printed handouts

Prerequisite: None, but general mathematical experience up to the level of elementary algebra and calculus is expected.

Course description: The course provides a classical introduction to a discipline rich in interesting solved and unsolved problems, some dating back to ancient times. Number theory is an area that requires a large variety of tricks and ideas to solve its problems. This is a course going deep into the beauties of this subject often called the Queen of Mathematics. The beginning of the course corresponds to an introductory level and it becomes more demanding as we proceed. We conclude with an outlook to certain aspects of advanced number theory. The lectures are accompanied with a large collection of problems of varying difficulty. Every second class is a problem solving class, where we solve problems individually and in groups. The main goal of the course is not only to know the strict material but also master the several problem solving techniques and strategies.

Topics:

Basic notions, divisibility, greatest common divisor, least common multiple, Euclidean algorithm, infinity of primes, congruences, residue systems, unique factorization.

Congruences, Euler's function f(n), Euler--Fermat Theorem, linear and quadratic congruences, Chinese Remainder Theorem, primitive roots modulo p, congruences of higher degree, power residues, very special cases of Dirichlet's theorem.

Quadratic residues, sums of two or four squares, Legendre-symbol and its properties, quadratic reciprocity, quadratic forms, applications (in computer science, cases of Dirichet's theorem).

Arithmetic functions. Multiplicativity and additivity, explicit formulae for f(n), d(n), and s(n), ``Valley Theorem'' and average number of dividors, perfect numbers.

Diophantine equations: linear equation, Pythagorean triplets, Fermat's Last Theorem, representation as sum of squares, some typical methods for solving Diophantine equations.

This and that. How many trees can you see in the woods, RSA cryptosystem, how to toss a coin via telephone, 13 proofs of the infinity of the primes, etc. A selection of theorems where number theory is applied.