On the exponential Diophantine equation Fnx ± Fmx = a with a ∈{Fr,Lr}

Abstract

In this paper, we will answer the question of when the sum or the difference of [Formula: see text]th powers of any two Fibonacci numbers becomes a Fibonacci number or a Lucas number. We prove that if this is possible with [Formula: see text], then [Formula: see text] may be only 3,4,5, or 10. Also, we give the result that the Diophantine equation in the title with [Formula: see text] has no solutions for [Formula: see text]