Tribonacci Numbers That Are Products of Two Lucas Numbers

Let $T_k$ be the $k^{\text{th}}$ Tribonacci number and $L_n$ be the $n^{\text{th}}$ Lucas number defined by their respective recurrence relation $T_k=T_{k-1}+T_{k-2}+T_{k-3}$ and $L_n=L_{n-1}+L_{n-2}$. In this study, we solve the Diophantine equation $T_k=L_m{L}_n$ for positive integer unknowns $m$, $n$, and $k$ and prove our results.