On the Diophantine equations of the form $\lambda_1U_{n_1} + \lambda_2U_{n_2} +\ldots + \lambda_kU_{n_k} = wp_1^{z_1}p_2^{z_2} \cdots p_s^{z_s}$

TL;DR

This paper establishes an effective upper bound for solutions to a class of Diophantine equations involving linear recurrence sequences and prime power products, extending previous bounds using linear forms in logarithms.

Contribution

It provides a new method to bound solutions of complex Diophantine equations involving linear recurrences and prime powers, generalizing prior results.

Findings

Existence of an effectively computable upper bound on solutions.

Extension of bounds for linear forms in logarithms to more general equations.

Application of the method to non-degenerate linear recurrence sequences.

Abstract

In this paper, we consider the Diophantine equation $\lambda_1U_{n_1}+\ldots+\lambda_kU_{n_k}=wp_1^{z_1} \cdots p_s^{z_s},$ where $\{U_n\}_{n\geq 0}$ 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).