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

14 April 2020, Version 1
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

Let $ \{F_{n}\}_{n\geq 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\geq 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.

Keywords

Fibonacci number
Pell equation
Linear form in logarithms
Reduction method

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