Instructor:  Dr. Mátyás DOMOKOS

Text:  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, unique factorization.

Congruences, residue systems, Euler's f(n) function, Euler-Fermat Theorem, Wilson's Theorem, linear congruences, Chinese Remainder Theorem, primitive roots modulo p, congruences of higher degree, power residues.

Quadratic residues,  quadratic congruences, sums of two squares, Legendre symbol and its properties, quadratic reciprocity.

Public key cryptography, evaluating exponentials, factoring integers, Mersenne primes.

Arithmetical functions. multiplicativity and additivity, Möbius function, explicit formulae for f(n), d(n), s(n), , mean value of f(n), perfect numbers, Fermat primes.

Special cases of Dirichlet's theorem, roots of unity, cyclotomic polynomials, primitive roots revisited..

Diophantine equations, linear equation, Pythagorean triples, special cases of Fermat's Last Theorem, some typical methods for solving Diophantine equations..

Prime numbers, elementary estimates, Chebyshev's Theorem.

Gauss integers, sums of two squares.

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