On the Skolem Problem and Prime Powers

TL;DR

This paper investigates a specialized version of the Skolem Problem, focusing on whether zeroes occur at indices of the form n=lp^k with bounded l,k and prime p, expanding understanding of recurrence zeros.

Contribution

It introduces a new variant of the Skolem Problem involving prime power indices with bounded parameters, providing insights into its decidability and complexity.

Findings

Characterization of the problem for specific bounds

Decidability results for certain parameter ranges

Connections to prime number distributions

Abstract

The Skolem Problem asks, given a linear recurrence sequence $(u_n)$, whether there exists $n\in\mathbb{N}$ such that $u_n=0$. In this paper we consider the following specialisation of the problem: given in addition $c\in\mathbb{N}$, determine whether there exists $n\in\mathbb{N}$ of the form $n=lp^k$, with $k,l\leq c$ and $p$ any prime number, such that $u_n=0$.