Related papers: Some conjectural continued fractions
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.
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'.
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…
In this small paper we bring together various open problems on geometric multidimensional continued fractions.
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…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and…
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization…
There are infinite processes (matrix products, continued fractions, $(r,s)$-matrix continued fractions, recurrence sequences) which, under certain circumstances, do not converge but instead diverge in a very predictable way. We give a…
This paper is a short survey of the recent results on examples of periodic two-dimensional continued fractions (in Klein's model). In the last part of this paper we formulate some questions, problems and conjectures on geometrical…
In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
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…
In this article, we will discover some new generalized identity regarding continued fractions. We will connect the results to Fibonacci numbers and Lucas numbers. For all the proof, we will use induction.
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…
Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
We present an improved incremental selection algorithm of the selection algorithm presented in [1] and prove all the selected conjectures.
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…