On the nonnegative integer solutions to the equation $F_n^2+F_m=2^a$

Abstract

The Diophantine equation $F_n^x+F_m^y=2^a,$ where $F_k$ denotes the $k$-th Fibonacci number and $n,m,a\in {\mathbb{Z}}_{\ge 0}$, has attracted attention in recent years. The cases $(x,y)=(1,1)$ and $(x,y)=(2,2)$ were completely resolved by Bravo and Luca, and by Jeong and Park, respectively. In this note, we investigate the mixed exponent case $(x,y)=(2,1)$ and we completely determine all nonnegative integer solutions.

arXiv comment12 pages