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

Abstract

For an integer k ≥ 2, let {F (k) n } n 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 x 2 −dy 2 = ±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.

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

“…(17) Furthermore, k ≤ n, for if not, we would then get that α n+1 ≤ δ n+1 ≤ δ k < α n+1 , a contradiction. Besides, given that k 1 < k 2 , we have by (6) and 16that…”

“…. In this paper, we let U := {F n + F m : n ≥ m ≥ 0} be the sequence of sums of two Fibonacci numbers. The first few members of U are U = {0, 1, 2, 3, 4, 5,6,7,8,9,10,11,13,14,15,16,18,21,22,23,24,26,29,34,35, . .…”

“…Several other related problems have been studied where x k belongs to some interesting positive integer sequences. For example, see [2,5,6,7,9,11,12,13,14,15].…”

On the $x$--coordinates of Pell equations that are products of two Lucas numbers

“…There are many other researchers who have studied related problems involving the intersection sequence {x n } n≥1 with linear recurrence sequences of interest. For example, see [4,8,7,9,12,13,14,16,17,20].…”

On the x-coordinates of Pell equations that are sums of two Padovan numbers

“…Several other related problems have been studied where x k belongs to some interesting positive integer sequences. For example, see [8,9,11,12,[14][15][16]18].…”