On the square of Fibonacci and Lucas numbers of the form (22s - 1)x + (2s+1)y.

Let F_k be a Fibonacci number and let L_k be a Lucas number. By applying Catalan's conjecture and the modular arithmetic method, we solve the exponential Diophantine equations of the form (2^2s-1)^x + (2^s+1)^y = F_2k and (2^2s - 1)^x + (2^s+1)^y = L²_k where x, y, k are non-negative integers and s is a positive integer. [ABSTRACT FROM AUTHOR]