On the x--coordinates of Pell equations which are k--generalized Fibonacci numbers

2018/03/28 by Ddamulira, Mahadi, Luca, Florian
#11A25 #11B39 #11J86 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1803.10434

Abstract

For an integer k≥ 2, let \F^(k)_n\_{n\geqslant 2-k} be the k--generalized Fibonacci sequence which starts with 0, …, 0,1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, for an integer d≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x²-dy² =± 1 which is a k--generalized Fibonacci number, with a couple of parametric exceptions which we completely characterise. This paper extends previous work from [17] for the case k=2 and [16] for the case k=3.

Related

• On the X-coordinates of Pell equations which are Tribonacci numbers
• On Pell numbers representable as product of two generalized Fibonacci numbers
• Positive Integer Solutions of the Pell Equation x²-dy²=N, % d∈ \k²± 4,k²± 1\ and N∈ \± 1,± 4\
• The Generalized Fibonacci and Lucas Solutions of The Pell Equations x²-(a²b²-b)y²=N and x²-(a²b²-2b)y²=N
• On integer values of the generating functions for sequences given by the Pell's equations