Common terms of generalized Pell and Narayana's cows sequences

2023/07/08 by Tripathy, Bibhu Prasad, Patel, Bijan Kumar
#11B39 #11J86 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2307.03919

Abstract

For an integer k ≥ 2, let \ P_n^(k) \_n be the k-generalized Pell sequence which starts with 0, …,0,1(k terms) and each term afterwards is the sum of k preceding terms. In this paper, we find all the solutions of the Diophantine equation P_n^(k) = N_m in non-negative integers (n, k, m) with k ≥ 2, where \ N_m \_m is the Narayana's cows sequence. Our approach utilizes the lower bounds for linear forms in logarithms of algebraic numbers established by Matveev, along with key insights from the theory of continued fractions.

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