Square-Classes in Lucas Sequences Having Odd Parameters

✍ Scribed by Wayne L. McDaniel; Paulo Ribenboim

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 258 KB
• Volume: 73
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2280

✦ Synopsis

Two or more terms of a sequence are said to be in the same square-class if the squarefree parts of the terms are identical. Let [U n (P, Q)] and [V n (P, Q)] denote the Lucas sequence and companion Lucas sequence, respectively, with parameters P and Q. For all odd relatively prime values of P and Q with discriminant P 2 &4Q>0, we show that [U n (P, Q)] and [V n (P, Q)] have only finitely many non-trivial square-classes and each square-class contains at most three terms. The square-classes are explicitly determined in ``most'' case, and an effectively computable bound on the number of square-classes, depending on P and Q, is obtained in the remaining cases.

1998 Academic Press

1. Introduction

The terms of a sequence [u n ] may be partitioned into disjoint classes by means of the following equivalence relation: u m tu n iff there exist non-zero integers x and y such that x 2 u m = y 2 u n , or equivalently, u m u n is a square. If u m tu n , u m and u n are said to be in the same square-class, and we write u m u n =g. A square-class containing more than one term of the sequence is called non-trivial.'' Let P and Q be non-zero integers, and : and ; (:>;) be the roots of X 2 &PX+Q=0. The Lucas sequence [U n ] and associated Lucas sequence'' [V n ] are defined, for n 0, by U n =U n (P, Q)=(: n &; n )Â(:&;), and V n =V n (P, Q)=: n +; n , respectively. In a recent paper [10], the authors proved that if P and Q are relatively prime integers with discriminant P 2 &4Q>0, each square-class of each sequence is finite and its Article No. NT982280