Instructor:  Dr. Mátyás DOMOKOS

Text:  Selected chapters of Melvyn B. Nathanson: Elementary Methods in Number Theory or Ivan Niven, Herbert S. Nathanson: An introduction to the theory of numbers
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 an introduction to a discipline rich in interesting solved and
unsolved problems, some dating back to very ancient times. This is a course going deep into the beauties of this wonderful
subject. The beginning of the course corresponds to an introductory level but 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. Some effort is devoted to master the techniques of strict mathematical reasoning.

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.

Quadratic residues,  sums of two or four squares, Legendre-symbol and its properties, quadratic reciprocity.

Computational problems, evaluating exponentials, compositeness tests, factoring integers, quadratic congruences, public key cryptography.

Arithmetical functions. Multiplicativity and additivity,   convolution, Möbius inversion formulae, mean values.

Prime numbers, elementary estimates, Mertens' theorems, the number of prime factors, the prime number theorem.

Dirichlet's theorem, real Dirichlet's series, characters, outline of the proof.

Remark. If the number of students registering for the introductory number theory courses will be below 15 we'll join the two number theory  courses.