Solving Skolem problem for negative indexed k-generalized Pell numbers

2025/09/02 by Mohapatra, Monalisa, Bhoi, Pritam Kumar, Panda, Gopal Krishna
#11D61 #FOS: Mathematics #Number Theory (math.NT) #Primary 11B39 #Secondary 11J86

paper · doi:10.48550/arxiv.2509.04503

Abstract

In this paper, we address the Skolem problem for the k-Generalized Pell numbers (P_n^(k)) extended to negative indices. We focus on identifying and bounding the indices n<0 for which P_n^(k)=0. In particular, we establish that the zero multiplicity is χ_k = 1 + (k(k-2)/4), if k is even and 1 + ((k - 1)²/4), if k is odd for all k ∈ [4, 500].

Citations

• On the k-generalized Fibonacci numbers with negative indices
• Powers of two as sums of two k-Fibonacci numbers
• Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers
• The zero multiplicity of linear recurrence sequences
• A generalization of a theorem of Baker and Davenport

Related

• The negative Pell equation
• Identities for the k-generalized Fibonacci sequence with negative indices and its zero-multiplicity
• Some properties of generalized k-Pell sequences
• On the k-generalized Fibonacci numbers with negative indices
• On the Largest Prime factor of the k-generalized Pell numbers