Related papers: Continued fractions of cubic irrationalities
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author described periods and sometimes precise form of…
We prove that the formula giving the exact value of the irrationality exponent of regular continued fractions remains valid for semi-regular continued fractions satisfyiong certain conditions.
In this paper we present some results related to the problem of finding periodic representations for algebraic numbers. In particular, we analyze the problem for cubic irrationalities. We show an interesting relationship between the…
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…
Several conjectural continued fractions found with the help of various algorithms are published in this paper.
In this small paper we bring together various open problems on geometric multidimensional continued fractions.
Continued fractions are used to give an alternate proof of $e^{x/y}$ is irrational.
In an earlier paper we introduced the notion of 'bifurcating continued fractions' in a heuristic manner. In this paper a formal theory is developed for the 'bifurcating continued fractions'.
We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…
A well known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two…
We show several properties related to the structure of the family of classes of two-dimensional periodic continued fractions. This approach to the study of the family of classes of nonequivalent two dimexsional periodic continued fractions…
Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of $p$--adic continued fractions, i.e. continued…
We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued…
The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…
Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and…
Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum…
This paper is devoted to a detailed exposition of geometry of continued fractions. We pay particular interest to the case of quadratic irrationalities and use the technique described to prove a criterion for the continued fraction of a…
In this paper we establish properties of independence for the continued fraction expansions of two algebraic numbers. Roughly speaking, if the continued fraction expansions of two irrational algebraic numbers have the same long sub-word,…