Squares in Euler triples from Fibonacci and Lucas numbers

Abstract

In this paper we shall continue to study from [4], for k = −1 and k = 5, the infinite sequences of triples A = (F_2n+1, F_2n+3, F_2n+5), B = (F_2n+1, 5F_2n+3, F_2n+5), C = (L_2n+1, L_2n+3, L_2n+5), D = (L_2n+1, 5L_2n+3, L_2n+5) with the property that the product of any two different components of them increased by k are squares. The sequences A and B are built from the Fibonacci numbers F_n while the sequences C and D from the Lucas numbers L_n. We show some interesting properties of these sequences that give various methods how to get squares from them.