Instructor: Dr. Csaba Szabóc

Text: M.B. Nathanson: Elementary Methods in Number Theory (Some further theorems are discussed, as well,  and also in some cases the proofs presented differ from
those in the book.)

Prerequisite: nothing

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. In the beginning of the course the methods correspond to an
introductory level.  The difficulty of  the problems    increases, and we conclude  with an outlook to certain aspects of advanced number theory.

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 Dirchlet's theorem.

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

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

Mersenne- and Fermat-primes. Form of possible prime divisors and  primality  tests  for Mersenne- and Fermat-numbers.

Arithmetical functions. Multiplicativity and additivity. f(n), d(n), ``Valley Theorem'' and average order, s(n), perfect numbers,  m(n),
elementary estimates of the number of primes.

Algebraic and transcendental numbers, algebraic integers,  Gaussian  integers, Cyclotomic polynomials unique factorization of polynomials. Complex numbers, roots of unity. Properties, irreducibility of cyclotomic polynomials. Primitive root revisited, special cases of Dirchlet's theorem.