The a, b problem extended to generalized Fibonacci and Lucas
Joseph Zaks, University of Haifa
The following a, b problem appeared first in the Intn' Math Olympiad competition a few years ago: "Show that if a and b are positive integers, such that the quantity t, defined by t=(a2+b2)/(ab+1) is an integer, then t is a square". Using my solution to this problem, I have derived the complete solution to the Diophantine equation x2 + y2 = z(xy+1). One can extend this approach to special cases of generalized Fibonacci and Lucas numbers, obtaining a result of the following form: for each value of n and for each even value of k, [F_(n-1)k]2 + [F_nk]2 = [L_nk][F_(n-1)k]F_nk +[F_k]2.