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

Abstract

Let $r\ge 1$ be an integer and $\mathbf{U}:=\{U_n{\}}_{n\ge 0}$ be the Lucas sequence given by $U_0=0,U_1=1$, and $U_{n+2}=r{U}_{n+1}+U_n$ for $n\ge 0$. In this paper, we explain how to find all the solutions of the Diophantine equation, $A{U}_n+B{U}_m=C{U}_{n_1}+D{U}_{m_1}$, in integers $r\ge 1$, $0\le m<n,0\le m_1<n_1$, $A{U}_n\ne C{U}_{n_1}$, 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.

8 pages, Accepted Manuscript