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

2025/04/26 by Batte, Herbert
#11B39 #11D45 #11D61 #11Y50 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.19052

Abstract

Let k ≥ 2 be an integer and consider the k-generalized Pell sequence \P_n^(k)\_{n ≥ 2-k}, defined by the initial values 0, …, 0, 0, 1 (a total of k terms), and the recurrence P_n^(k) = 2P_n-1^(k) + P_n-2^(k) + ⋯ + P_n-k^(k), for all n≥ 2. For any integer m, let P(m) denote the largest prime factor of m, with the convention P(0) = P(±1) = 1. In this paper, we prove that for n ≥ 4, the inequality 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.

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