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Solving Skolem problem for negative indexed $k-$generalized Pell numbers

Monalisa Mohapatra, Pritam Kumar Bhoi, Gopal Krishna Panda

Abstract

In this paper, we address the Skolem problem for the $k$-generalized Pell sequence $(P_n^{(k)})_{n\geq2-k}$ 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 of $P_n^{(k)}$ is $ χ_k = \lfloor k^2/4\rfloor$ for all $k \in [4, 500].$

Solving Skolem problem for negative indexed $k-$generalized Pell numbers

Abstract

In this paper, we address the Skolem problem for the -generalized Pell sequence extended to negative indices. We focus on identifying and bounding the indices for which In particular, we establish that the zero multiplicity of is for all

Paper Structure

This paper contains 5 sections, 2 theorems, 79 equations, 1 table.

Table of Contents

  1. Introduction
  2. Auxiliary Results
  3. k-Generalized Pell Sequence
  4. Proof of Theorem \ref{['main result 1']}
  5. Proof of Corollary \ref{['coro 1.2']}

Key Result

Theorem 1.1

The largest nonnegative integer solution $n$ to the Diophantine equation $P_{-n}^{(k)} = 0$ satisfies the following bounds:

Theorems & Definitions (2)

  • Theorem 1.1
  • Theorem 2.1