On the Largest Prime factor of the $k$-generalized Pell numbers

TL;DR

This work analyzes the arithmetic of the $k$-generalized Pell numbers $P_n^{(k)}$ by establishing an explicit lower bound for the largest prime factor: $\mathcal{P}(P_n^{(k)})>\dfrac{1}{104}\log\log n$ for all $n\ge 4$ and $k\ge 2$. The authors combine a Binet-type representation, properties of the dominant root $\alpha$ of the characteristic polynomial, and bounds for linear forms in logarithms (Matveev) with LLL lattice reduction to convert prime-factor information into effective bounds on $n$ and $k$, including a case split depending on the relative size of $n$ and $k$. They also determine all $(n,k)$ for which $P_n^{(k)}$ has largest prime factor at most $7$, finding exactly four instances: $P_4^{(2)}=12$, $P_6^{(2)}=70$, $P_6^{(3)}=84$, and $P_{10}^{(5)}=4116$. The results advance our understanding of prime divisors in linear recurrences and demonstrate an effective combination of Baker-type bounds and lattice reduction with computer verification for complete classification.

Abstract

Let $k \ge 2$ be an integer and consider the $k$-generalized Pell sequence $\{P_n^{(k)}\}_{n \ge 2-k}$, defined by the initial values $0, \ldots, 0, 0, 1$ (a total of $k$ terms), and the recurrence $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \cdots + P_{n-k}^{(k)}$, for all $n\ge 2$. For any integer $m$, let $\mathcal{P}(m)$ denote the largest prime factor of $m$, with the convention $\mathcal{P}(0) = \mathcal{P}(\pm1) = 1$. In this paper, we prove that for $n \ge 4$, the inequality $\mathcal{P}(P_n^{(k)}) > (1/104) \log \log n$ holds. Additionally, we find all $k$-generalized Pell numbers $P_n^{(k)}$, whose largest prime factor does not exceed $7$.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.1: Matveev, MAT
• Definition 2.2
• Lemma 2.1
• Lemma 2.2: Lemma VI.1 in SMA
• Lemma 2.3: Lemma 7 in GL
• Lemma 3.1
• proof