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Top Papers in Continued fraction analysis

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The Algorithm of Shor for Prime Factorization

Continued Fractions and Probability Estimations in the Shor Algorithm -- A Detailed and Self-Contained Treatise

We present the relevant results and proofs from the theory of continued fractions in detail (even in more detail than in text books) filling the gap to allow a complete comprehension of the algorithm of shor for prime factorization.

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Octagonal continued fraction and diagonal changes

In this short note we show that the octagon Farey map introduced by Smillie
and Ulcigrai is an acceleration of the diagonal changes algorithm introduced by
Delecroix and Ulcigrai.

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A Case Study of branched-continued-fraction representations of ratios of hypergeometric series and type II multiples on the step-line

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

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Continued Fraction approach to Gauss Reduction Theory

Jordan Normal Forms serve as excellent representatives of conjugacy classes
of matrices over closed fields. Once we knows normal forms, we can compute
functions of matrices, their main invariant, etc.

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Linear fractional transformations and non-linear leaping convergents of some continued fractions

For $\alpha_0 = \left[a_0, a_1, \ldots\right]$ an infinite continued fraction
and $\sigma$ a linear fractional transformation, we study the continued
fraction expansion of $\sigma(\alpha_0)$ and its c

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Continued Fractions and Factoring

Legendre found that the continued fraction expansion of $\sqrt N$ having odd
period leads directly to an explicit representation of $N$ as the sum of two
squares. Similarly, it is shown here that the

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Invariant densities for random continued fractions

We continue the study of random continued fraction expansions, generated by
random application of the Gauss and the R\'enyi backward continued fraction
maps. We show that this random dynamical system

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Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding

Repeatedly folding a strip of paper in half and unfolding it in straight
angles produces a fractal: the dragon curve. Shallit, van der Poorten and
others showed that the sequence of right and left tur

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Farey-subgraphs and Continued Fractions

In this note, we study a family of subgraphs of the Farey graph, denoted as
$\mathcal{F}_N$ for every $N\in\mathbb{N}.$
We show that $\mathcal{F}_N$ is connected if and only if $N$ is either equal
to

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A New Continued Fraction Expansion and Weber's Class Number Problem

We give a new continued fraction expansion algorithm for certain real numbers
related to Weber's class number problem. By considering the analogy of the
solution of Pell's equations, we get an explici

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On integer partitions and continued fraction type algorithms

We show that the additive-slow-Farey version of the traditional continued
fractions algorithm has a natural interpretation as a method for producing
integer partitions of a positive number $n$ into tw

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Generalized continued fraction expansions for $π$ and $e$

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 sta

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On the Borel complexity of continued fraction normal, absolutely abnormal numbers

We show that normality for continued fractions expansions and normality for
base-$b$ expansions are maximally logically separate. In particular, the set of
numbers that are normal with respect to the

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Simple Continued Fractions an approach for High School students

This paper introduces high school students to continued fractions and
develops basic properties of Finite Continued Fractions and Infinite Continued
Fractions. This also includes computation of the qu

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How to prove Ramanujan's $q$-continued fractions

By using Euler's approach of using Euclid's algorithm to expand a power
series into a continued fraction, we show how to derive Ramanujan's
$q$-continued fractions in a systematic manner.

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Two complementary relations for the Rogers-Ramanujan continued fraction

Let $R(q)$ be the Rogers-Ramanujan continued fraction. We give different
proofs of two complementary relations for $R(q)$ given by Ramanujan and proved
by Watson and Ramanathan. Our proofs only use pr

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Elementary proofs of generalized continued fraction formulae for $e$

In this short note we prove two elegant generalized continued fraction
formulae $$e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\ddots}}}}$$
and $$e= 3+\cfrac{-1}{4+\cfrac{-2}{5+\cfrac{-3}{6+\c

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Geometric algorithms for the notion continued

Diagonal changes for surfaces in hyperelliptic components

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A Central Limit Theorem for Rosen Continued Fractions

We prove a central limit theorem for Birkho? sums of the Rosen continued
fraction algorithm. A Lasota-Yorke bound is obtained for general
one-dimensional continued fractions with the bounded variation

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kaleidoscopic and self-similar aspects of the integral Apollonian gaskets

Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets

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