scholarly journals Period-length equality for the nearest integer and nearest square continued fraction expansions of a quadratic surd

2011 ◽  
Vol 46 (2) ◽  
pp. 269-282 ◽  
Author(s):  
Keith R. Matthews ◽  
John P. Robertson
Keyword(s):  
Continued Fraction ◽  
Period Length ◽  
Quadratic Surd ◽  
Continued Fraction Expansions

scholarly journals On certain continued fraction expansions of fixed period length

1999 ◽  
Vol 89 (1) ◽  
pp. 23-35 ◽  
Author(s):  
A. van der Poorten ◽  
H. Williams
Keyword(s):  
Continued Fraction ◽  
Period Length ◽  
Continued Fraction Expansions ◽  
Fixed Period

scholarly journals Some Criteria for Class Numbers to Be Non-One

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Ahmad Issa ◽  
Hasan Sankari
Keyword(s):  
Positive Integer ◽  
Continued Fraction ◽  
Period Length ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Pell Equation ◽  
Fixed Integer ◽  
Real Quadratic Fields ◽  
Continued Fraction Expansions ◽  
Real Quadratic

Let d be a positive integer which is not a perfect square and n be any nonzero fixed integer. Then, the equation x 2 − d y 2 = n is known as the general Pell equation. In this paper, we give some criteria for class numbers of certain real quadratic fields to be greater than one, depending on the solvability of the general Pell equation, ideals in quadratic orders, and the period length of the simple continued fraction expansions of d .

scholarly journals CLASSIFICATION AND SYMMETRIES OF A FAMILY OF CONTINUED FRACTIONS WITH BOUNDED PERIOD LENGTH

2012 ◽  
Vol 93 (1-2) ◽  
pp. 53-76
Author(s):  
K. H. F. CHENG ◽  
R. K. GUY ◽  
R. SCHEIDLER ◽  
H. C. WILLIAMS
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Period Length ◽  
Continued Fraction Expansion ◽  
Quadratic Irrational ◽  
Continued Fraction Expansions

AbstractIt is well known that the regular continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a family of quadratics known as Schinzel sleepers. This paper provides a method for generating every Schinzel sleeper and investigates their period lengths as well as both their horizontal and vertical symmetries.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

Sign in / Sign up

Export Citation Format

Share Document