The Product of Like-Indexed Terms in Binary Recurrences Original Research Article

In recent work by Hajdu and Szalay, Diophantine equations of the form (ak−1)(bk−1)=x2 were completely solved for a few pairs (a, b). In this paper, a general finiteness theorem for equations of the form ukvk=xn is proved, where uk and vk are terms in certain types of binary recurrence sequences. Also, a unified computational approach for solving equations of the type (ak−1)(bk−1)=x2 is described, and this approach is used to completely solve such equations for almost all (a,b) in the range 1