*The a, b problem extended to generalized Fibonacci and Lucas
sequences*

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=(a^{2}+b^{2})/(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 x^{2} + y^{2} = 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}.