Related papers: Continued fraction algorithms and Lagrange's theor…
We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P^1R, determines a…
Rational approximations to a square root $\sqrt{k}$ can be produced by iterating the transformation $f(x) = (dx+k)/(x+d)$ starting from $\infty$ for any positive integer $d$. We show that these approximations coincide infinitely often with…
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…
In this paper we define "a continued fraction expansion of the exponential integral $E_{1}(x)$ at infinity", which is analogous to the regular continued fraction expansion of real numbers, and prove that this expansion gives the same…
The paper examines the structure of the periodic continued fraction for $\sqrt{d}$ and gives formulae for the central term as well as the repeated partial quotients occurring in its period.
We establish an equidistribution result for push-forwards of certain locally finite algebraic measures in the adelic extension of the space of lattices in the plane. As an application of our analysis we obtain new results regarding the…
We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general…
Several conjectural continued fractions found with the help of various algorithms are published in this paper.
We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not.
Gosper developed an algorithm for performing arithmetic operations on continued fractions (CFs), getting a CF as the result. Straightforward implementation of the algorithm leads to infinite loops on some inputs. Here we present a modified…
The direct calculation of the Generalized operator entropy proves difficult by the appearance of rational exponents of matrices. The main motivation of this work is to overcome these difficulties and to present a practical and efficient…
We build, for real quadratic fields, infinitely many periodic continuous fractions uniformly bounded, with a seemingly better bound than the known ones. We do that using continuous fraction expansions with the same shape as those of real…
In this paper we give an algorithm to calculate the coefficients of the p-adic expansion of a rational numbers, and we give a method to decide whether this expansion is periodic or ultimately periodic.
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,…
In this paper, we improve some transcendence results for $p$--adic continued fractions. In particular, we prove that palindromic and quasi--periodic $p$--adic continued fractions converge either to transcendental numbers or quadratic…
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,…
Motivated by the optimal continued fractions studied independently by Selenius and Bosma, we define and introduce algorithms producing superoptimal continued fraction expansions of irrationals. The convergents of these expansions…
Let $f(x)$ be a square free quartic polynomial defined over a quadratic field $K$ such that its leading coefficient is a square. If the continued fraction expansion of $\displaystyle \sqrt{f(x)}$ is periodic, then its period $n$ lies in the…
Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…
We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.