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

Published 17 May 2026 in math.NT | (2605.17699v1)

Abstract: We study multiplicative dependence between terms of the $k$ -generalized Pell sequence $(P_n{(k)})_{n\ge 2-k}$ , defined by the linear recurrence [ P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}, ] with initial conditions $P_0{(k)} = \dots = P_{-(k-2)}{(k)} = 0$ and $P_1{(k)} = 1$ . For $k\ge 2$ we determine all pairs $(m,n)$ with $n>m\ge 0$ such that $P_n{(k)}$ and $P_m{(k)}$ are multiplicatively dependent. The main result states that the only solutions occur for very small $k,m,n$ (which are listed explicitly). The proof uses lower bounds for linear forms in logarithms (Matveev), the Baker-Davenport reduction algorithm, and a computational search.

Abstract PDF Upgrade to Chat

Authors (4)

Summary

The paper classifies all index pairs (m, n) for which k-generalized Pell numbers exhibit multiplicative dependence, showing that beyond trivial cases in the initial segment, no nontrivial dependencies exist.
It employs advanced Diophantine techniques including Matveev’s theorem, a Binet-type formula, and the Baker-Davenport reduction to derive sharp bounds on potential solutions.
The findings reinforce the arithmetic rigidity of recurrence sequences by proving that for k ≥ 3, multiplicative independence holds once indices exceed the initial power-of-two segment.

Multiplicative Independence in the Sequence of $k$ -Generalized Pell Numbers

Introduction

The paper addresses the characterization of multiplicative dependence between terms of the $k$ -generalized Pell sequence, denoted $(P_n^{(k)})_{n \geq 2-k}$ , which is defined via the linear recurrence $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ with initial conditions $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ , $P_1^{(k)} = 1$ . Multiplicative dependence for integers $a,b \neq 0$ is defined as the existence of nonzero integers $x, y$ such that $a^x = b^y$ .

Building on prior results for $k$ -generalized Fibonacci and Lucas numbers, the authors provide a complete classification of all index pairs $k$ 0 with $k$ 1 such that $k$ 2 and $k$ 3 are multiplicatively dependent. The approach synthesizes techniques from Diophantine approximation, including Matveev's lower bounds for linear forms in logarithms, the Baker-Davenport reduction (in the Dujella-Pethő form), and exhaustive computational search.

Background

The analysis is rooted in the structural properties of the $k$ 4-generalized Pell sequence, which generalizes the well-known properties of classical sequences such as the Fibonacci, Lucas, and Pell numbers. For $k$ 5, the sequence recovers the standard Pell numbers. The study of multiplicative dependence between sequence elements generalizes classical problems in the arithmetic of linear recurrence sequences, following foundational work by Carmichael and more recent full classifications for $k$ 6-generalized Fibonacci and Lucas numbers.

The dominant root $k$ 7 of the characteristic polynomial $k$ 8 plays a crucial role, as does the function $k$ 9 appearing in the Binet-type formula for $(P_n^{(k)})_{n \geq 2-k}$ 0. Estimates for these quantities, and for their associated logarithmic heights, are critical in the analytic bounds used throughout.

Main Results

The principal theorem of the paper asserts: For $(P_n^{(k)})_{n \geq 2-k}$ 1, if $(P_n^{(k)})_{n \geq 2-k}$ 2 and $(P_n^{(k)})_{n \geq 2-k}$ 3 and $(P_n^{(k)})_{n \geq 2-k}$ 4 are multiplicatively dependent, then either $(P_n^{(k)})_{n \geq 2-k}$ 5 (trivial), $(P_n^{(k)})_{n \geq 2-k}$ 6 (where terms are exact powers of two), or $(P_n^{(k)})_{n \geq 2-k}$ 7 (the classical Pell sequence exception). No other pairs exist.

Thus, apart from initial segments linked to explicit powers of two and a single exception for $(P_n^{(k)})_{n \geq 2-k}$ 8, all pairs of distinct $(P_n^{(k)})_{n \geq 2-k}$ 9-Pell numbers are multiplicatively independent as soon as one index exceeds $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 0.

Methodology

The proof combines several advanced methods in Diophantine analysis:

Initial Segment Analysis: For $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 1, $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 2 and the problem reduces to analyzing multiplicative dependence among powers of two, which is trivial. The exception $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 3 is inherited from the classical structure of Pell numbers.
Linear Forms in Logarithms: For larger $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 4 and $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 5, the proof employs the Binet-type formula to approximate $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 6. Forming a linear form in logarithms of algebraic numbers, the inequality $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 7 is established, where $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 8, and $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}$ 9 relate to the exponents and indices of the sequence.
Matveev's Theorem: Explicit lower bounds on nonzero linear forms in logarithms are derived using Matveev's powerful theorem. This produces initial polynomial bounds in $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 0 for possible solutions to the central Diophantine equation.
Baker-Davenport Reduction: Because bounds from Matveev's theorem are too coarse for exhaustive computation, a single iteration of the Baker-Davenport reduction, as implemented by Dujella and Pethő, sharpens the upper bound to $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 1 for all $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 2.
Asymptotic and Exhaustive Analysis: For $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 3, asymptotic expansions (Cooper-Howard formula) and integrality conditions show that no nontrivial solutions can arise. For $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 4, a finite exhaustive search verifies that no further solutions exist beyond those in the initial segment.

Numerical and Constructive Aspects

The authors stress that for all $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 5, the upper bound $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 6 reduces the set of candidate pairs to a finite, computationally tractable set. An explicit computer search falsifies the existence of any exceptional or hidden multiplicative dependencies beyond the classified cases.

For $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 7, analytic estimation leveraging the structure of the dominant root and the relative size of correction terms guarantees integrality constraints are only satisfied trivially, precluding new solutions for large $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 8.

Theoretical and Practical Implications

The result establishes that the $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ 9-generalized Pell numbers, for $P_1^{(k)} = 1$ 0, display multiplicative independence among their elements outside the initial power-of-two segment. This properties mirror and extend those known for $P_1^{(k)} = 1$ 1-generalized Fibonacci and Lucas numbers, enriching the understanding of arithmetic dynamics within linear recurrence sequences.

The work's approach also showcases the efficacy of combining transcendence theory, continued fractions, and computational methods in resolving family-wide Diophantine equations. The technical toolkit—especially Matveev's theorem and the Baker-Davenport reduction—proves robust for managing both small and large parameter regimes.

In terms of future directions, the methods exemplified may be adapted to other classes of linear recurrences (including those with non-integer coefficients or more general initial conditions), or to questions of additive and combinatorial independence among sequence elements. The near-total absence of nontrivial multiplicative dependence for $P_1^{(k)} = 1$ 2 underscores a form of algebraic rigidity which may have further consequences in the study of unlikely intersections and arithmetic dynamical systems.

Conclusion

This paper provides a definitive classification of all cases of multiplicative dependence between distinct terms in the $P_1^{(k)} = 1$ 3-generalized Pell sequence. The result—excluding trivial and initial power-of-two cases—shows that for $P_1^{(k)} = 1$ 4, $P_1^{(k)} = 1$ 5 and $P_1^{(k)} = 1$ 6 are multiplicatively independent once $P_1^{(k)} = 1$ 7. The methodology successfully integrates analytic, reduction, and computational techniques, establishing a framework with prospective application to analogous Diophantine finiteness problems.