On the $x$-coordinates of Pell equations which are Fibonacci numbers II

Abstract

For an integer d ≥ 2 which is not a square, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − dy 2 = ±4 which is a Fibonacci number, except when d = 2, 5, cases in which we have exactly two values of x being members of the Fibonacci sequence.

The X -coordinates of Pell equations and Padovan numbers

“…Besides the trivial case n 1 = 1 (for both equations), which implies X 1 = P l1 , the only nontrivial solutions are (n 1 , l 1 , X 1 ) = (2, 9, 2) and (n 1 , l 1 , X 1 ) = (2,16,5), in the first case which leads to (d, However, since we assume that l 2 > 200, we get a contradiction, so l 2 ≤ 200 leading to n 2 < 64, 8 . Checking the last range we only obtained the following possibilities: X 1 = 2 = P 4 = P 5 , X 2 = 7 = P 9 , with d = 3, and X 1 = 5 = P 8 , X 2 = 49 = P 16 , with d = 6, and X 1 = 1 = P 1 = P 2 = P 3 , X 2 = 3 = P 6 X 3 = 7 = P 9 , with d = 2, and X 1 = 2 = P 4 = P 5 , X 2 = 9 = P 10 , with d = 5, respectively.…”

“…In the literature, there are many papers investigating for which d there are members of the sequence {X n } n≥1 or {Y m } m≥1 belonging to some interesting sequences of positive integers such as the sequence of all base 10-repdigits [2], the sequence of all base b-repdigits [4], the sequence of Fibonacci numbers [5,8], and the sequence of Tribonacci numbers [7]. For most sequences, one expects that the answer to such a question has at most one positive integer solution n for any given d except maybe for a few (finitely many) values of d .…”

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

“…. 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].…”

X-coordinates of Pell equations which are Lucas numbers

“…For each of these two values of d, the equation X n ∈ U has two solutions n, see also [6]. In [11], it was shown that if U is the sequence of Fibonacci numbers, then the equation X n ∈ U has at most one positive integer solution n, except when d = 2 for which there are exactly two solutions, see also [8]. In [12], it was shown that if U = T is the sequence of Tribonacci numbers given by T 0 = 0, T 1 = T 2 = 1 and T n+3 = T n+2 + T n+1 + T n for all n ≥ 0, then the equation X n ∈ U has at most one positive integer solution n with a few exceptions in d which were completely determined.…”