de Théorie des Nombres de Bordeaux 16 ( 2004

Let P,Q be positive, relatively prime and odd integers such that P 2 − 4Q > 0. We study the sequences (xn)n>0 of positive integers satisfying the recursion formula xn+1 = Pxn − Qxn−1. They generalize the classical Lucas sequences (Un(P,Q)) and (Vn(P,Q)). The prime divisors of Vn(P,Q) for n = 3 · 2 have nice properties which, through the computation of the Legendre Symbols of suitable xn’s modulo these primes, give an efficient method for trying to find all squares (also double squares, triple squares, ...) in the sequence (xn). This is applied to Diophantine equations of the form x − Ey = k, x − Ey = k when E is the squarefree part of an integer P 2 − 4, P odd. We construct sequences (xn) containing squares with arbitrarily large indices. We also show how to find sequences (xn) containing three squares.