On the $X$-coordinates of Pell equations which are Tribonacci numbers

Abstract

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 = ±1 which is a Tribonacci number, with a few exceptions that we completely characterize.

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

“…Returning back to (12) and (14) and using the fact that (x 1 , y 1 ) is the smallest positive solution to the Pell equation 1, we obtain…”

“…. 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 which are k–generalized Fibonacci numbers

“…They proved that equation (1.3) has at most one solution (n, m) in positive integers except for d = 2, in which case equation (1.3) has the three solutions (n, m) = (1, 1), (1,2), (2,4). Luca, Montejano, Szalay and Togbé [16] considered the Diophantine equation (1.4) x n = T m , where {T m } m 0 is the sequence of Tribonacci numbers given by T 0 = 0, T 1 = 1, T 2 = 1 and T m+3 = T m+2 + T m+1 + T m for all m 0. They proved that equation (1.4) has at most one solution (n, m) in positive integers for all d except for d = 2 when equation (1.4) has the three solutions (n, m) = (1, 1), (1,2), (3,5) and when d = 3 case in which equation (1.4) has the two solutions (n, m) = (1, 3), (2,5).…”

X-coordinates of Pell equations which are Lucas numbers

“…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. In this paper, we consider the same problem for the sequence U = (L m ) m≥0 of Lucas numbers, defined as L 0 = 2, L 1 = 1 and L m+2 = L m+1 + L m , for m ≥ 0.…”