Effective Resolution of Diophantine equations of the form $u_n+u_m=w{p}_1^{z_1}\cdots p_s^{z_s}$

Abstract

Let $u_n$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_1,p_2,\dots ,p_s$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $u_n+u_m=w{p}_1^{z_1}\cdots p_s^{z_s}$ and prove under mild technical restrictions effective finiteness results. In particular we give explicit upper bounds for $n,m$ and $z_1,\dots ,z_s$. Furthermore, we provide a rather efficient algorithm to solve Diophantine equations of the described type and we demonstrate our method by an example.