Linearly periodic continued fractions

2021 ◽  
Vol 105 (564) ◽  
pp. 442-449
Author(s):  
Kantaphon Kuhapatanakul ◽  
Lalitphat Sukruan
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Positive Integers ◽  
Periodic Continued Fractions

An infinite simple continued fraction representation of a real number α is in the form $$\eqalign{& {a_0} + {1 \over {{a_1} + {1 \over {{a_2} + {1 \over {{a_3} + {1 \over {}}}}}}}} \cr & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ddots \cr} $$ where $${a_0}$$ is an integer, and $${a_i}$$ are positive integers for $$i \ge 1$$. This is often written more compactly in one of the following ways: $${a_0} + {1 \over {{a_1} + }}{1 \over {{a_2} + }}{1 \over {{a_3} + }} \ldots \;{\rm{or}}\;\left[ {{a_0};\;{a_1},\;{a_2},\;{a_3} \ldots } \right]$$ .

scholarly journals A Note on Continued Fractions

1960 ◽  
Vol 12 ◽  
pp. 303-308 ◽  
Author(s):  
A. Oppenheim
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Quadratic Irrational ◽  
Image Position ◽  
Positive Integers ◽  
Periodic Continued Fractions

Any real number y leads to a continued fraction of the type(1)where ai, bi are integers which satisfy the inequalities(2)by means of the algorithm(3)the a's being assigned positive integers. The process terminates for rational y; the last denominator bk satisfying bk ≥ ak + 1. For irrational y, the process does not terminate. For a preassigned set of numerators ai ≥ 1, this C.F. development of y is unique; its value being y.Bankier and Leighton (1) call such fractions (1), which satisfy (2), proper continued fractions. Among other questions, they studied the problem of expanding quadratic surds in periodic continued fractions. They state that “it is well-known that not only does every periodic regular continued fraction represent a quadratic irrational, but the regular continued fraction expansion of a quadratic irrational is periodic.

scholarly journals Non-periodic continued fractions for quadratic irrationalities

2016 ◽  
Vol 12 (05) ◽  
pp. 1329-1344
Author(s):  
Michael O. Oyengo
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Geometric Series ◽  
Quadratic Irrational ◽  
Number Formula ◽  
Periodic Continued Fractions

A well-known theorem of Lagrange states that the simple continued fraction of a real number [Formula: see text] is periodic if and only if [Formula: see text] is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two interlacing geometric series and show that in certain cases they converge to quadratic irrationalities. This phenomenon is connected with certain sequences of polynomials whose properties we examine further.

scholarly journals On Absolutely Normal and Continued Fraction Normal Numbers

2018 ◽  
Vol 2019 (19) ◽  
pp. 6136-6161 ◽  
Author(s):  
Verónica Becher ◽  
Sergio A Yuhjtman
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Main Difficulty ◽  
Normal Numbers ◽  
Continued Fraction Expansion ◽  
Mathematical Operations ◽  
Construction Works

Abstract We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n4 mathematical operations. The construction works by defining successive refinements of appropriate subintervals to achieve, in the limit, simple normality to all integer bases and continued fraction normality. The main difficulty is to control the length of these subintervals. To achieve this we adapt and combine known metric theorems on continued fractions and on expansions in integers bases.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

scholarly journals Application of Wynn's epsilon algorithm to periodic continued fractions

1987 ◽  
Vol 30 (2) ◽  
pp. 295-299 ◽  
Author(s):  
M. J. Jamieson
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Quadratic Surd ◽  
Partial Quotients ◽  
Infinite Continued Fraction ◽  
Image Position ◽  
Epsilon Algorithm ◽  
Periodic Continued Fractions

The infinite continued fractionin whichis periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

scholarly journals Limits of unbounded sequences of continued fractions

1990 ◽  
Vol 41 (3) ◽  
pp. 509-512
Author(s):  
Jingcheng Tong
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Quotients ◽  
Limit Points ◽  
Necessary And Sufficient ◽  
Positive Integers

Let X = {xk}k≥1 be a sequence of positive integers. Let Qk = [O;xk,xk−1,…,x1] be the finite continued fraction with partial quotients xi(1 ≤ i ≤ k). Denote the set of the limit points of the sequence {Qk}k≥1 by Λ(X). In this note a necessary and sufficient condition is given for Λ(X) to contain no rational numbers other than zero.

Some properties of the continued fraction expansion of (m/n) e1/q

1970 ◽  
Vol 67 (1) ◽  
pp. 67-74 ◽  
Author(s):  
K. R. Matthews ◽  
R. F. C. Walters
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Positive Integers

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.

ON THE INCREASING PARTIAL QUOTIENTS OF CONTINUED FRACTIONS OF POINTS IN THE PLANE

2021 ◽  
pp. 1-8
Author(s):  
MEIYING LÜ ◽  
ZHENLIANG ZHANG
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Large N ◽  
Partial Quotients ◽  
Positive Integers

Abstract For any x in $[0,1)$ , let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be its continued fraction. Let $\psi :\mathbb {N}\to \mathbb {R}^+$ be such that $\psi (n) \to \infty $ as $n\to \infty $ . For any positive integers s and t, we study the set $$ \begin{align*}E(\psi)=\{(x,y)\in [0,1)^2: \max\{a_{sn}(x), a_{tn}(y)\}\ge \psi(n) \ {\text{for all sufficiently large}}\ n\in \mathbb{N}\} \end{align*} $$ and determine its Hausdorff dimension.

scholarly journals Mixed periodic Jacobi continued fractions

1986 ◽  
Vol 104 ◽  
pp. 129-148 ◽  
Author(s):  
Yoshifumi Kato
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Jacobi Matrix ◽  
Padé Approximant ◽  
Positive Real ◽  
Pade Approximant

Let b0 be a positive real number andbe a Jacobi matrix. We can associate with them a Jacobi continued fraction, which will be abbreviated to a J fraction from the next section, as followswhere An(z)/Bn(z) is the n-th Padé approximant of φ(z).

scholarly journals Cluster algebras and continued fractions

2017 ◽  
Vol 154 (3) ◽  
pp. 565-593 ◽  
Author(s):  
İlke Çanakçı ◽  
Ralf Schiffler
Keyword(s):  
Asymptotic Behavior ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Cluster Algebra ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Periodic Continued Fractions

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

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