The Euclidean Algorithm and Simple Continued Fractions

2008 ◽  
Keyword(s):  
Continued Fractions ◽  
Euclidean Algorithm

Microcomputer-assisted Discoveries: Euclidean Algorithm and Continued Fractions

1983 ◽  
Vol 76 (7) ◽  
pp. 510-548
Author(s):  
Clark Kimberling
Keyword(s):  
Continued Fractions ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Positive Integers

Students can use microcomputers to cut through algorithms and computations to gain mathematical insights. This approach is especially true for the Euclidean algorithm, so often used to find the greatest common divisor (GCD) of two positive integers. The Euclidean algorithm also yields continued fractions, at least far enough for students to find patterns and discover truths about numbers.

scholarly journals CONTINUED FRACTIONS WITH ODD PARTIAL QUOTIENTS

2012 ◽  
Vol 08 (06) ◽  
pp. 1541-1556 ◽  
Author(s):  
E. N. ZHABITSKAYA
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Continued Fractions ◽  
Functional Equations ◽  
Continued Fraction ◽  
Euclidean Algorithm ◽  
Partial Quotients ◽  
Limit Distribution Function

Every Euclidean algorithm is associated with a kind of continued fraction representation of a number. The representation associated with "odd" Euclidean algorithm we will call "odd" continued fraction. We consider the limit distribution function F(x) for sequences of rationals with bounded sum of partial quotients for "odd" continued fractions. In this paper we prove certain properties of the function F(x). Particularly this function is singular and satisfies a number of functional equations. We also show that the value F(x) can be expressed in terms of partial quotients of the "odd" continued fraction representation of a number x.

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.

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