Continued Fractions

  title={Continued Fractions},
  author={Alain Lasjaunias},
The study of continued fractions is an ancient part of elementary Number Theory. It was studied by Leonhard Euler in the 18-th century. Actually, a remarkable paper from him was translated from Latin language into English and published thirty years ago [1]. The subject has been treated very deeply by Oskar Perron at the beginning of the 20 th century, in a famous book which has been edited several times [2]. It can also be found in several books on Number Theory, among them a famous one is “An… 
On the periodicity of an algorithm for p-adic continued fractions
In this paper we study the properties of an algorithm (introduced in [6]) for generating continued fractions in the field of p–adic numbers Qp. First of all, we obtain an analogue of the Galois’
On Continued Fractions
This paper on the geometry, algebra and arithmetics of continued fractions is based on a lecture for students, teachers and a non-specialist audience, beginning with the history of the golden number
Intermediate convergents and a metric theorem of Khinchin
A landmark theorem in the metric theory of continued fractions begins this way: Select a non‐negative real function f defined on the positive integers and a real number x, and form the partial sums
Prime Numbers and the Convergents of a Continued Fraction
Continued fractions offer a concrete representation for arbitrary real numbers. The continued fraction expansion of a real number is an alternative to the representation of such a number as a
Orderings of the rationals and dynamical systems
A class of one-dimensional maps is introduced which can be used to generate the binary trees in dierent ways and study their ergodic properties and some random processes arising in a natural way in this context.
Continued fractions in non-Euclidean imaginary quadratic fields.
In the Euclidean imaginary quadratic fields, continued fractions have been used to give rational approximations to complex numbers since the late 19th century. A variety of algorithms have been
On Consecutive 1’s in Continued Fractions Expansions of Square Roots of Prime Numbers
Abstract In this note, we study the problem of existence of sequences of consecutive 1’s in the periodic part of the continued fractions expansions of square roots of primes. We prove unconditionally


A Survey of Diophantine Approximationin Fields of Power Series
Abstract. The fields of power series (or perhaps better called formal numbers) are analogues of the field of real numbers. Many questions in number theory which have been studied in the setting of
Diophantine Approximation and Continued Fraction Expansions of Algebraic Power Series in Positive Characteristic
Abstract In a recent paper M. Buck and D. Robbins have given the continued fraction expansion of an algebraic power series when the base field is F 3. We study its rational approximation property in
An Introduction to the Theory of Numbers
  • E. T.
  • Mathematics
  • 1946
THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.
On continued fractions and diophantine approximation in power series fields
1. Continued fractions in fields of series. While some deep work has been done on continued fractions in power series fields, there does not seem to exist a general introduction, or an easily
An Introduction to the Theory of Numbers
This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,
Rational approximation to algebraic numbers of small height: the Diophantine equation
Following an approach originally due to Mahler and sharpened by Chud- novsky, we develop an explicit version of the multi-dimensional ''hypergeometric method'' for rational and algebraic
The Continued Fraction Expansion of An Algebraic Power Series Satisfying A Quartic Equation
Abstract Some time ago Mills and Robbins (1986, J. Number Theory 23, No. 3, 388-404) conjectured a simple closed form for the continued fraction expansion of the power series solution ƒ = a 1 x −1 +
Continued fractions for certain algebraic power series
Badly approximable power series in characteristic 2
1. Statement of results We let F be the finite field with two elements, and let K be the field F((x')) of all formal power series in x-' over F. Every element of K not in the field of rational
Mathematical Thought from Ancient to Modern Times
This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation