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