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

Mahadi Ddamulira (1) , Florian Luca (2)

1 TU Graz - Graz University of Technology [Graz]
2 UNAM - Instituto de Matematicas

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.

Mots clés: Fibonacci number; Pell equation; linear form in logarithm; reduction method

Domaines: Théorie des nombres [math.NT]

Mahadi Ddamulira, Florian Luca. The $x-$coordinates of Pell equations and sums of two Fibonacci numbers II. Proceedings Mathematical Sciences, In press. ⟨hal-02266438⟩