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

Abstract

Let $S:=\{p_1,\dots ,p_{\ell }\}$ 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 $(U_n{)}_{n\ge 0}$ 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 $a{U}_n+b{U}_m=z_1+\cdots +z_r,$ where $n\ge m$ and $z_1,\dots ,z_r\in {\mathcal{U}}_S$.

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