The x-coordinates of Pell equations and sums of two Fibonacci numbers II

Abstract

Let $$ \{F_{n}\}_{n\ge 0} $$ be the sequence of Fibonacci numbers defined by $$ F_0=0 $$ , $$ F_1 =1$$ and $$ F_{n+2}= F_{n+1} +F_n$$ for all $$ n\ge 0 $$ . In this paper, for an integer $$ d\ge 2 $$ which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation $$ x^2-dy^2=\pm 4 $$ which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize.