Squares in Lucas Sequences with Rational Roots

In 1997, Darmon and Merel proved the stunning result that the Diophantine equation x + y = z has no nontrivial integer solutions for n ≥ 4. This can be interpreted as saying that if {vn} represents a Lucas sequence of the second kind, defined by a quadratic polynomial with rational roots, then the equation vn = x , with x an integer, implies that n ≤ 3. The goal of the present paper is to prove a similar, but partial, result for the case that the sequence is a Lucas sequence of the first kind, whose defining polynomial has rational roots. In other words, our goal is to study the Diophantine equation z = (x − yn)/(x− y). Employing a combination of recently proved Diophantine results of Bennett and Skinner, Poonen, Darmon and Merel, Wiles, and Ribet, from the modularity approach, together with recent advances by N. Bruin on the effective Chabauty method for determining all rational points on a given curve of genus greater than one, we deduce that the equation in question is not solvable for a substantial proportion of positive integer values n.