*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 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, very
special cases of Dirichlet's theorem.

**Quadratic residues, **
sums of two or four squares, Legendre-symboland its properties, quadratic
reciprocity, quadratic forms.

**Arithmetical functions.**
Multiplicativity and additivity, explicit formulae for *f(n)*,*d(n)*, and *s(n), *``Valley Theorem'' and average order,
perfect numbers.

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

**Algebraic** and **transcendental **numbers, algebraic
integers, Gaussian integers, complex numbers, roots of
unity.

Cyclotomic polynomials**, ** unique factorization of
polynomials.Primitive root revisited, some more cases of Dirchlet's
theorem.

Quadratic number fields, unique
factorization.