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

1 Correction to: Periodica Mathematica Hungarica https://doi.org/10.1007/s10998-017-0226-8

Abstract

In this work, we correct an oversight from [1].

2 Introduction

For a positive squarefree positive integer d and the Pell equation $X^2-dY^2 = \pm \,1$, where $X,Y \in \mathbb {Z}^{+}$, it is well known that all its solutions (X, Y) have the form $X+Y{\sqrt{d}}=X_k+ Y_k{\sqrt{d}}= (X_1 + Y_1{\sqrt{d}})^k$ for some $k \in \mathbb {Z}^{+}$, where $(X_{1},Y_{1})$ is the smallest positive integer solution. Let $\{T_n\}_{n\ge 0}$ be the Tribonacci sequence given by $T_0=0,~T_1 = T_2 = 1,~T_{n+3} = T_{n+2}+T_{n+1}+T_n$ for all $n\ge 0$. Let $U=\{T_n+T_m: n\ge m\ge 0\}$ be the set of non-negative integers which are sums of two Tribonacci numbers. In [1], we looked at Pell equations $X^2-dY^2 = \pm \,1$ such that the containment $X_\ell \in U$ has at least two positive integer solutions $\ell$. The following result was proved.

Theorem 1.1

For each squarefree integer d, there is at most one positive integer $\ell$ such that $X_\ell \in U$ except for $d\in \{2, 3, 5, 15, 26\}$.

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

The following intermediate result is Lemma 4.1 in [1].

Lemma 1.2

Let $(m_i, n_i, \ell _i)$ be two solutions of $T_{m_i} + T_{n_i} = X_{\ell _i}$, with $0 \le m_i < n_i$ for $i=1,2$ and $1 \le \ell _1 < \ell _2$. Then

$$\begin{aligned} m_1< n_1 \le 1535, \quad \ell _1 \le 1070 \quad \mathrm{and} \quad n_2 < 2.5\cdot 10^{42}. \end{aligned}$$

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

$$\begin{aligned} X_{\ell _1}=F_{n_1}+F_{m_1},\quad \ell _1\in [1,1070]\quad 2\le m_1<n_1\le 1535. \end{aligned}$$

Unfortunately, the case $\ell _1=1$ was omitted in [1]. Here, we discuss the missed case $\ell _1=1$.

In order to reduce the above bound on $n_2$ from Lemma 1.2, we do not consider the equation $P_{\ell _1}^{\pm }(X_1) = X_1$ since there is no polynomial equation to solve; instead, we consider each minimal solution $\delta :=\delta (X_1, \epsilon )$ of Pell equation $X^2-dY^2 = \epsilon = \pm \,1$, for each $X_{1}= T_{m_{1}}+T_{n_{1}}$, according to the bounds in Lemma 1.2. Thus, after some reductions using the Baker–Davenport method on the linear form in logarithms $\varGamma _1$ and $\varGamma _2$ from [1, inequalities 3.9 and 3.12], for $(m,n,\ell )=(m_2,n_2,\ell _2)$, one shows that the only range for the variables to be considered is

$$\begin{aligned} \ell _1 = 1, \quad 1 \le m_1< n_1 \le 1811, \quad 1 \le m_2 < n_2 \le 3210, \quad \mathrm{and} \quad 2 \le \ell _2 \le 2220.\qquad \end{aligned}$$

(1)

Now, with this new bound on $n_2$, by the same procedure (LLL algorithm and continued fractions) used on the linear form in logarithms $\varGamma _3$, $\varGamma _4$ and $\varGamma _5$ in [1, inequalities 3.15–3.26], we reduce again the bound on $n_1$ given in Lemma 1.2. Then, further cycles of reductions (for $n_2$ with the new bound of $n_1$) on $\varGamma _1$ and $\varGamma _2$ yield the following result.

Lemma 1.3

Let $(m_i, n_i, \ell _i)$ be two solutions of $T_{m_i} + T_{n_i} = X_{\ell _i}$, with $0 \le m_i < n_i$ for $i=1,2$. If $\ell _1 =1$, then $1 \le m_1 < n_1 \le 160$, $1 \le m_2< n_2 < 250$ and $2 \le \ell _2 \le 175$.

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.