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O.Forster: Selected Topics from Number Theory


im SS 2024
am Mathematischen Institut der LMU München
Mi 14-16, A027; Beginn 17.04.2024

Übungen dazu 14-tägig, Mi 16-18, A027; Beginn 24.04.2024

Klausur am 17. Juli 2024, 14 hct, A027

Contents:   This course deals mainly with

   Continued Fractions and Diophantine Approximations

A regular continued fraction has the form
  
which we abbreviate by x = cfrac(a_0,a_1,a_2,...,a_n,...). Every real number x admits such a CF (continued fraction) expansion with integers a_n, where a_n >= 1 for n >= 1. This expansion converges in the sense that the so called convergents x_n := cfrac(a_0,a_1,...,a_n) converge to x for n to infinity. Whereas the ordinary decimal expansion of a rational number is periodic, the CF expansion of a rational number is finite and quadratic irrationals have a periodic CF expansion. For example sqrt(2) = cfrac(1,2,2,2,...), sqrt(3) = cfrac(1,1,2,1,2,1,2,1,2,..) and cfrac(1,1,1,1,1,1,...) is the famous golden ratio (1+sqrt(5))/2.
Some interesting CF expansions for the Euler number e and the number pi = 3.14159... are 

   e = cfrac(2,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14,1,1,16,...)
  pi = cfrac(3,7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,2,1,...)

The first formula was proven by Euler, while the CF expansion for pi is completely mysterious, no pattern is known.
The convergents of the CF expansion of an irrational number x are rational numbers p/q, which give a good approximation of x, namely
   |x - p/q| < 1/q^2.  
On the other hand, a classical theorem of Liouville asserts that for every algebraic number x of degree n > 1 there exists c > 0 such that
   |x - p/q| > c/q^n
for all rationals p/q. This enabled Liouville to construct the first explicit examples of transcendental numbers. The exponent n in Liouville's estimate was lowered to 2 + eps for every eps > 0 by Fields medalist F.K.Roth, which is best possible.

Continued fractions are also useful to solve Pell's equation   x^2 - Dy^2 = 1  
and to determine the units in real quadratic number fields.

Language:  If everybody in the audience understands German, the course will be given in German:
   Ausgewählte Kapitel aus der Zahlentheorie
   Kettenbrüche und diophantische Approximationen

Prerequisites: Basic notions of algebra, number theory and analysis

ECTS:   3 points
Masterstudiengang Mathematik PO 2021: WP 13, 16, 40, 41
Masterstudiengang Mathematik PO 2011: WP 18.1, 15.2, 17.2

Gliederung

  1. Einführung. Eindeutigkeit und Konvergenz
  2. Approximations-Eigenschaften von Kettenbrüchen
  3. Äquivalente Zahlen
  4. Periodische Kettenbrüche
  5. Faktorisierung mit Kettenbrüchen
  6. Pellsche Gleichung und die Einheiten reell-quadratischer Zahlkörper
  7. Kettenbruch-Entwicklung der Euler-Zahl e = exp(1).
  8. Irrationalitäs- und Transzendenz-Beweise für e und pi.

Literature

  • J. Borwein, A. van der Poorten, J. Shallit, W. Zudilin: Neverending Fractions. Cambridge University Press 2014
  • A. Baker: Transcendental Number Theory. Cambridge Mathematical Library 2022
  • O. Forster: Algorithmische Zahlentheorie. 2.Aufl. Springer Spektrum 2015
  • A.Ya. Khinchin: Continued Fractions. Dover 1997 (Nachdruck der engl. Übersetzung der 3. russ. Auflage)
  • O. Perron: Die Lehre von den Kettenbrüchen, Bd I, II. Teubner 1954/1957

Vorlesungen vergangener Semester

Bücher/Books    Eprints    Software
Otto Forster 2024-02-28