Number theory 2

Mathematics BSc
Course description
2013.

Number theory 2

Hours lect+pc	Credits lect+pc	Assessment	Specialization	Course code lect/pc	Semester	Status
2 + 0	2 + 0	exam	pure math.	mm1c1se4m	4	optional

Course coordinator

Strong	Weak	Prerequisites
Lecture
		Strong: Algebra3P-p (mm1c2al3m)
		Strong: Analysis2L (mm1c1an2) or Foundations of AnalysisL (mm1c1ap2)

Remarks

Syllabus designed by:

Course objectives

We illustrate the structure of modern number theory.

Literature

G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. Online edition.

Syllabus

Elements of analytic number theory and prime number theory, Definition of the Riemann zeta function for s>1, its Euler product representation, the existance of infiniely many primes. Large gaps between primes. Dirichlet theorem on primes in arithmetic progressions (without proof), special cases.
Diophantine equations. The equation x²+y²=n, Gaussian integers. Other quadratic extensions, there exists a quadratic extension without unique prime factorization. Representation as the sum of 3, resp 4 squares, Lagrange's theorem. Waring's problem, g(k), G(k), lower bounds for them. Pell equations.
Diophantine approximation, Dirichlet's approximation theorem.
Elements of algebraic number theory. Algebraic and transcendental numbers, Liouville's theorem, the construction of a transcendental number.
Elements of combinatorial number theory, Sidon sets. The method of generating functions, the Fibonacci numbers. The elements of the geometry of numbers, the circle problem. The estimate of ∑ 1/p (p prime).