On the Shorey-Tijdeman Diophantine equation involving terms of Lucas sequences

TL;DR

This paper develops a method to find all integer solutions to a Shorey-Tijdeman type Diophantine equation involving Lucas sequences, considering parameters like sequence index and sequence parameters.

Contribution

It provides a comprehensive approach to solve a class of Diophantine equations involving Lucas sequences with unknown sequence parameters.

Findings

All solutions for the Diophantine equation are characterized.

The method applies to any integer r ≥ 1 and given coefficients.

Explicit solutions are obtained for specific cases.

Abstract

Let $r\ge 1$ be an integer and ${\bf U}:=\{U_n\}_{n\ge 0}$ be the Lucas sequence given by $U_0=0,~U_1=1$, and $U_{n+2}=rU_{n+1}+U_n$ for $n\ge 0$. In this paper, we explain how to find all the solutions of the Diophantine equation, $AU_{n}+BU_{m}=CU_{n_1}+DU_{m_1}$, in integers $r\ge 1$, $0\le m<n,~0\le m_1<n_1$, $AU_n\ne CU_{n_1}$, where $A,B,C,D$ are given integers with $A\ne 0,~B\ne 0$, $m,n,m_1,n_1$ are nonnegative integer unknowns and $r$ is also unknown.