X-coordinates of Pell equations as sums of two tribonacci numbers

Abstract

In this paper, we find all positive squarefree integers d such that the Pell equation X 2 − dY 2 = ±1 has at least two positive integer solutions (X, Y ) and (X , Y ) such that both X and X are sums of two Tribonacci numbers.

Correction to: X-coordinates of Pell equations as sums of two Tribonacci numbers

“…Furthermore, for each d ∈ {2, 3, 5, 15, 26}, all solutions to X ∈ U were given together with the representations of these X 's as sums of two Tribonacci numbers. Unfortunately, there was an oversight in [1], which we now correct.…”

“…An exhaustive search in this last range finds no new solutions. Hence, albeit the work in [1] missed one branch of computations which are described in this note, this does not affect the final result Theorem 1.1.…”

“…Let U = {T n + T m : n ≥ m ≥ 0} be the set of non-negative integers which are sums of two Tribonacci numbers. In [1], we looked at Pell equations X 2 − dY 2 = ± 1 such that the containment X ∈ U has at least two positive integer solutions . The following result was proved.…”

“…The rest of the argument in [1] were just reductions of the above parameters. The first step of the reduction consisted in finding all the solutions to…”

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

“…c Indian Academy of Sciences 1 has infinitely many positive integer solutions (x, y). By putting (x 1 , y 1 ) for the smallest positive solutions to (1), all solutions are of the forms (x k , y k ) for some positive integer k, where…”