Prerequisite: None, but general mathematical experience up to the level of elementary algebra and calculus is expected.
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.
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
Cyclotomic polynomials, unique factorization of polynomials.Primitive root revisited, some more cases of Dirchlet's theorem.
Quadratic number fields, unique factorization. Additional topics (if time permits): sum of two, four and three squares; proof of Dirichlet`s theorem on primes in arithmetic progressions.