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Resolution of the Skolem Problem for $k$-Generalized Lucas Sequences

Monalisa Mohapatra, Pritam Kumar Bhoi, Gopal Krishna Panda

Abstract

This paper provides a complete solution to Skolem's problem for the $k$-generalized Lucas sequence $(L_n^{(k)})_{n \in \mathbb{Z}}$ with a primary focus on its behavior at negative indices. We characterize the zero-distribution of this sequence by identifying and bounding all indices $n < 0$ such that $L_n^{(k)} = 0$. Our central result establishes that the zero-multiplicity $δ_k$ of the sequence is $(k-1)(k-2)/2$ for all $k.$

Resolution of the Skolem Problem for $k$-Generalized Lucas Sequences

Abstract

This paper provides a complete solution to Skolem's problem for the -generalized Lucas sequence with a primary focus on its behavior at negative indices. We characterize the zero-distribution of this sequence by identifying and bounding all indices such that . Our central result establishes that the zero-multiplicity of the sequence is for all
Paper Structure (12 sections, 4 theorems, 98 equations, 1 table)

This paper contains 12 sections, 4 theorems, 98 equations, 1 table.

Table of Contents

  1. Introduction
  2. Auxiliary Results
  3. Proof of Theorem \ref{['main result 1']}
  4. Proof of Theorem \ref{['coro 1.2']} for $k\in [4,500]$
  5. Matrix Representation and Structural Properties of $(Q_n)$
  6. Proof of Theorem \ref{['thm2']}
  7. Proof of Theorem \ref{['coro 1.2']} for $k>500$.
  8. Case 1 ($k$ is even):
  9. Case 2 (k is odd):
  10. A lower bound for $n$ in terms of $k$:
  11. An upper bound for $n$ in terms of $k$:
  12. Absolute bounds for n and k:

Key Result

Theorem 1.1

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

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1