TITLE:
Full Euclidean Algorithm by Means of a Steady Walk
AUTHORS:
Carlos M. Falcon Rodriguez, Maria A. Garcia Cruz, Claudia Falcon
KEYWORDS:
Extended Euclidean Algorithm, Greatest Common Divisor, Incommensurable Numbers, Steady Walk, Diophantine Equation
JOURNAL NAME:
Applied Mathematics,
Vol.12 No.4,
April
14,
2021
ABSTRACT: Let x and y be two positive real numbers with x y. Consider a traveler, on the interval [0, y/2], departing from 0 and taking steps of length equal to x. Every time a step reaches an endpoint of the interval, the traveler rebounds off the endpoint in order to complete the step length. We show that the footprints of the traveler are the output of a full Euclidean algorithm for x and y, whenever y/x is a rational number. In the case that y/x is irrational, the algorithm is, theoretically, not finite; however, it is a new tool for the study of its irrationality.