Sums of $S$-units and perfect powers in recurrence sequences

P. K. Bhoi, G. K. Panda, S. S. Rout

Published 2021-04-10Version 1

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 study the finiteness result for the solutions of the Diophantine equation $U_n + U_m = z_1 +\cdots +z_r,$ where $n\geq m$ and $z_1, \ldots, z_r\in \mathcal{U}_S$. We also study the Diophantine equation $U_n + U_m =wy^q$ with $w$ is a non-zero integer. In this case, we show that sum of two terms of recurrence sequence $(U_n)_{n\geq 0}$ does not contain any $q$-th power if $q$ is large enough.