Padovan and Perrin numbers as products of two generalized Lucas numbers

Abstract

Let $P_m$ and $E_m$ be the $m$-th Padovan and Perrin numbers respectively. Let $r, s$ be non-zero integers with $r\ge 1$ and $s\in \lbrace -1, 1\rbrace $, let $\lbrace U_n\rbrace _{n\ge 0}$ be the generalized Lucas sequence given by $U_{n+2}=rU_{n+1} + sU_n$, with $U_0=0$ and $U_1=1.$ In this paper, we give effective bounds for the solutions of the following Diophantine equations \[ P_m=U_nU_k\quad \text{and}\quad E_m=U_nU_k\,, \] where $m$, $ n$ and $k$ are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.