Full Euclidean Algorithm by Means of a Steady Walk  

作　　者：Carlos M. Falcon Rodriguez Maria A. Garcia Cruz Claudia Falcon Carlos M. Falcon Rodriguez;Maria A. Garcia Cruz;Claudia Falcon(Grupo de Investigación de Análisis Matemático, Instituto de Formación docente Salomé Ureña, Santo Domingo, D.N., República Dominicana;Wake Forest University, Winstom-Salem, NC, USA)

机构地区：[1]Grupo de Investigación de Análisis Matemático, Instituto de Formación docente Salomé Ureñ a, Santo Domingo, D.N., República Dominicana [2]Wake Forest University, Winstom-Salem, NC, USA

出　　处：《Applied Mathematics》2021年第4期269-279,共11页应用数学（英文）

摘　　要：Let <em>x</em> and <em>y</em> be two positive real numbers with <em>x</em> < <em>y</em>. Consider a traveler, on the interval [0, <em>y</em>/2], departing from 0 and taking steps of length equal to <em>x</em>. 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 <em>x</em> and <em>y</em>, whenever <em>y</em>/<em>x</em> is a rational number. In the case that <em>y</em>/<em>x</em> is irrational, the algorithm is, theoretically, not finite;however, it is a new tool for the study of its irrationality.Let <em>x</em> and <em>y</em> be two positive real numbers with <em>x</em> < <em>y</em>. Consider a traveler, on the interval [0, <em>y</em>/2], departing from 0 and taking steps of length equal to <em>x</em>. 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 <em>x</em> and <em>y</em>, whenever <em>y</em>/<em>x</em> is a rational number. In the case that <em>y</em>/<em>x</em> is irrational, the algorithm is, theoretically, not finite;however, it is a new tool for the study of its irrationality.

关 键 词：Extended Euclidean Algorithm Greatest Common Divisor Incommensurable Numbers Steady Walk Diophantine Equation 

分 类 号：TP3[自动化与计算机技术—计算机科学与技术]