Multiplicative independence in the sequence of $k$-generalized Pell numbers

Technical Summary: Multiplicative Independence in the Sequence of k-Generalized Pell Numbers

Problem Statement
This paper investigates the multiplicative dependence between terms of the $k$-generalized Pell sequence, denoted as $(P^{(k)}_n)_{n \ge 2-k}$. The sequence is defined by the linear recurrence:
$$P^{(k)}_n = 2P^{(k)}_{n-1} + P^{(k)}_{n-2} + \dots + P^{(k)}_{n-k}$$
with initial conditions $P^{(k)}_0 = \dots = P^{(k)}_{-(k-2)} = 0$ and $P^{(k)}_1 = 1$. For $k=2$, this reduces to the classical Pell sequence.

The central objective is to determine all pairs of indices $(m, n)$ with $n > m \ge 0$ and integers $k \ge 2$ such that the terms $P^{(k)}_n$ and $P^{(k)}_m$ are multiplicatively dependent. Two non-zero integers $a$ and $b$ are multiplicatively dependent if there exist non-zero integers $x, y$ such that $a^x = b^y$. The authors aim to solve the Diophantine equation:
$$(P^{(k)}_n)^x = (P^{(k)}_m)^y$$
excluding trivial cases where terms equal 1 or where the dependence arises solely from the initial segment of the sequence where terms are powers of two.

Methodology
The proof strategy follows the established framework used for $k$-generalized Fibonacci and Lucas numbers, combining analytic number theory with computational verification. The methodology proceeds in three main stages:

1. Analytic Approximation and Linear Forms in Logarithms:
   The authors utilize the Binet-type formula for $P^{(k)}_n$, which expresses the term as $P^{(k)}_n = g_k(\alpha)\alpha^n + e_n$, where $\alpha$ is the dominant root of the characteristic polynomial $\Psi_k(x) = x^k - 2x^{k-1} - \dots - 1$, and $g_k(\alpha)$ is a specific algebraic function.
   By substituting this into the multiplicative dependence equation, the problem is transformed into bounding a linear form in logarithms:
   $$\Lambda = (y-x)\log g_k(\alpha) + (nx - my)\log \alpha$$
   Using Matveev's theorem on lower bounds for linear forms in logarithms, the authors derive a polynomial upper bound for the index $n$ in terms of $k$. Specifically, for $k \ge 3$ and $n > m \ge k+1$, they establish:
   $$n < 6.2 \times 10^{33} k^8 (\log k)^6$$

2. Reduction via Baker-Davenport Algorithm:
   The initial bound derived from Matveev's theorem is too large for direct computation. To reduce this, the authors employ the Baker-Davenport reduction method (specifically the version by Dujella and Pethő). This involves analyzing the continued fraction expansion of the ratio $\tau_k = \frac{\log g_k(\alpha)}{\log \alpha}$.
   By utilizing properties of convergents and the Dujella-Pethő lemma, the authors reduce the upper bound for $n$ to a manageable constant independent of $k$ for the range $k \le 850$. Specifically, the bound is reduced to $n \le 218$ (conservatively taken as $n < 300$ for safety).

3. Case Separation and Asymptotic Analysis:
   The problem is split into two regimes based on the value of $k$:

   • Small $k$ ($k \le 850$): With the reduced bound $n < 300$, the authors perform an exhaustive computer search over all valid combinations of $k, m, n, x, y$.
   • Large $k$ ($k > 850$): For this range, the authors use an asymptotic argument. They exploit the fact that for large $k$, the dominant root $\alpha$ is extremely close to 2, and $P^{(k)}_n \approx 2^{n-1}$. By expanding the terms to the second order (using the Cooper-Howard expansion), they show that the equation $(P^{(k)}_n)^x = (P^{(k)}_m)^y$ implies $x=y$ and subsequently $n=m$, which contradicts the assumption $n > m$. Thus, no solutions exist for $k > 850$.

Key Contributions and Results
The paper provides a complete classification of multiplicatively dependent pairs in the $k$-generalized Pell sequence. The main result, Theorem 1.1, states that for $k \ge 2$ and $n > m \ge 2$, the terms $P^{(k)}_n$ and $P^{(k)}_m$ are multiplicatively dependent if and only if:

1. Trivial Case: $m=1$ (since $P^{(k)}_1 = 1$).
2. Power-of-Two Families: $2 \le m < n \le k+1$. In this initial segment, $P^{(k)}_n = 2^{n-1}$ and $P^{(k)}_m = 2^{m-1}$, leading to dependencies of the form $(2^{n-1})^x = (2^{m-1})^y$.
3. Exceptional Classical Case: $(k, n, m) = (2, 3, 0)$. Here, $P^{(2)}_3 = 2$ and $P^{(2)}_0 = 0$. (Note: The abstract mentions $P^{(k)}_0=0$ as a trivial case in the context of the equation, though strictly speaking, $0^x=0$ requires careful handling; the paper lists this as the only non-trivial solution involving the classical Pell sequence outside the power-of-two range).

The authors explicitly state that no other multiplicatively dependent pairs exist. For $k \ge 3$, the only non-trivial dependencies occur strictly within the initial segment where terms are exact powers of two.

Significance
The paper extends the study of multiplicative dependence from classical sequences (Fibonacci, Lucas) and $k$-generalized Fibonacci/Lucas sequences to the $k$-generalized Pell sequence. It demonstrates that the structural properties of the Pell sequence (specifically the behavior of its dominant root and the initial power-of-two segment) lead to a similar classification of solutions. The work confirms that the "richness" of solutions found in other sequences does not extend to the Pell sequence beyond the initial trivial segments and the specific classical case. The proof rigorously combines deep theoretical tools (Matveev's bounds, Baker-Davenport reduction) with computational verification to close the problem for all $k$.