*Instructor:* Dr. Csaba SZABÓ

*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**, ivisibility, 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.

**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.

**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.