Sums of $S$-units in sum of terms of recurrence sequences

TL;DR

This paper establishes a finiteness result for solutions to a Diophantine equation involving linear recurrence sequences and sums of $S$-units, where $S$ is a finite set of primes.

Contribution

It provides a new finiteness theorem for solutions of a specific Diophantine equation involving recurrence sequences and $S$-units.

Findings

Finiteness of solutions for the given Diophantine equation

Conditions on the recurrence sequence and $S$-units

Extension of previous results to more general sequences

Abstract

Let $S := \{p_1,\ldots ,p_{\ell}\}$ be a finite set of primes and denote by $\mathcal{U}_S$ the set of all rational integers whose prime factors are all in $S$. Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two. In this paper, we provide a finiteness result for the solutions of the Diophantine equation $aU_n + bU_m = z_1 +\cdots +z_r,$ where $n\geq m$ and $z_1, \ldots, z_r\in \mathcal{U}_S$.