On the Diophantine equations of the form ${\lambda }_1{U}_{n_1}+{\lambda }_2{U}_{n_2}+\dots +{\lambda }_k{U}_{n_k}=w{p}_1^{z_1}{p}_2^{z_2}\cdots p_s^{z_s}$

Abstract

In this paper, we consider the Diophantine equation ${\lambda }_1{U}_{n_1}+\dots +{\lambda }_k{U}_{n_k}=w{p}_1^{z_1}\cdots p_s^{z_s},$ where $\{U_n{\}}_{n\ge 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; $w$ is a fixed non-zero integer; $p_1,\dots ,p_s$ are fixed, distinct prime numbers; ${\lambda }_1,\dots ,{\lambda }_k$ are strictly positive integers; and $n_1,\dots ,n_k,z_1,\dots ,z_s$ are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions $(n_1,\dots ,n_k,z_1,\dots ,z_s)$. In our proof, we use lower bounds for linear forms in logarithms, extending the work of Pink and Ziegler (2016), Mazumdar and Rout (2019), Meher and Rout (2017), and Ziegler (2019).

14 pages, 1 table