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Observations on continued fractions from Ford's point of view

Hindov, Raul LU (2016) In Bachelor's Theses in Mathematical Sciences MATK01 20161
Mathematics (Faculty of Sciences)
Abstract (Swedish)
Detta är ett projekt på en geometrisk metod för att visualisera beräkning av kedjebråk, som har utvecklats av Lester R. Ford på 1930-talet.
Beskrivningen av metoden följs med användning tekniken av Ford cirklar på tre olika fall.
I motsats till ett kedjebråk med positiva heltal koefficienter, som alltid är konvergent, kan ett kedjebråk med reella koefficienter divergera.
Detta projekt ger en alternativ geometrisk bevis för en av de klassiska konvergens satser om kedjebråk.
Bland primtalsfaktorisering metoder finns det en som använder kedjebråk. Vi ger en alternativ geometrisk bevis för lemmat som är basen för metoden
och för en följdsats som reducerar factorisering problemet med en faktor av 2.
Kedjebråk är också anslutna till ämnet... (More)
Detta är ett projekt på en geometrisk metod för att visualisera beräkning av kedjebråk, som har utvecklats av Lester R. Ford på 1930-talet.
Beskrivningen av metoden följs med användning tekniken av Ford cirklar på tre olika fall.
I motsats till ett kedjebråk med positiva heltal koefficienter, som alltid är konvergent, kan ett kedjebråk med reella koefficienter divergera.
Detta projekt ger en alternativ geometrisk bevis för en av de klassiska konvergens satser om kedjebråk.
Bland primtalsfaktorisering metoder finns det en som använder kedjebråk. Vi ger en alternativ geometrisk bevis för lemmat som är basen för metoden
och för en följdsats som reducerar factorisering problemet med en faktor av 2.
Kedjebråk är också anslutna till ämnet Diophantine approximation, och projektet innebär en alternativ geometriska bevis för Dirichlets approximation sats.
Vi antar läsaren att vara en matematik student som har tagit en kurs om talteori och om analytiska funktioner, och har viss tid på händerna för att läsa projektet. (Less)
Abstract
This is a project on a geometric method to visualise the calculation of continued fractions, which was developed by Lester R. Ford in the 1930s. The description of the method is followed by the use of the technique of Ford circles on three distinct cases. In contrast to a continued fraction with positive integer coefficients, which is always convergent, a continued fraction with real coefficients might diverge. This project provides an alternative geometric proof for one of the classical convergence theorems on continued fractions.
Among the integer factorization methods there is one that uses continued fractions. We give an alternative geometric proof for the lemma that is the base for the method and for a corollary that reduces the... (More)
This is a project on a geometric method to visualise the calculation of continued fractions, which was developed by Lester R. Ford in the 1930s. The description of the method is followed by the use of the technique of Ford circles on three distinct cases. In contrast to a continued fraction with positive integer coefficients, which is always convergent, a continued fraction with real coefficients might diverge. This project provides an alternative geometric proof for one of the classical convergence theorems on continued fractions.
Among the integer factorization methods there is one that uses continued fractions. We give an alternative geometric proof for the lemma that is the base for the method and for a corollary that reduces the factoring problem by a factor of 2.
Continued fractions are also connected to the subject of Diophantine approximation, and the project presents an alternative geometric proof for the Dirichlet's approximation theorem.
We assume the reader to be a mathematics student who has taken a course on Number Theory and on Analytic Functions, and has certain amount of time on her hands to read the paper. (Less)
Please use this url to cite or link to this publication:
author
Hindov, Raul LU
supervisor
organization
course
MATK01 20161
year
type
M2 - Bachelor Degree
subject
keywords
continued fractions, Ford circles, integer factoring, Diophantine approximation
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4053-2016
ISSN
1654-6229
other publication id
2016:K13
language
English
id
8892215
date added to LUP
2017-02-08 15:58:31
date last changed
2017-02-08 15:58:31
@misc{8892215,
  abstract     = {This is a project on a geometric method to visualise the calculation of continued fractions, which was developed by Lester R. Ford in the 1930s. The description of the method is followed by the use of the technique of Ford circles on three distinct cases. In contrast to a continued fraction with positive integer coefficients, which is always convergent, a continued fraction with real coefficients might diverge. This project provides an alternative geometric proof for one of the classical convergence theorems on continued fractions.
Among the integer factorization methods there is one that uses continued fractions. We give an alternative geometric proof for the lemma that is the base for the method and for a corollary that reduces the factoring problem by a factor of 2.
Continued fractions are also connected to the subject of Diophantine approximation, and the project presents an alternative geometric proof for the Dirichlet's approximation theorem.
We assume the reader to be a mathematics student who has taken a course on Number Theory and on Analytic Functions, and has certain amount of time on her hands to read the paper.},
  author       = {Hindov, Raul},
  issn         = {1654-6229},
  keyword      = {continued fractions,Ford circles,integer factoring,Diophantine approximation},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor's Theses in Mathematical Sciences},
  title        = {Observations on continued fractions from Ford's point of view},
  year         = {2016},
}