On $k$-generalized Fibonacci numbers which are perfect powers of Lucas numbers

Let $(F_{n}^{(k)})$ and $(L_{n})$ be the $k$-generalized Fibonacci and Lucas sequences, respectively. In this paper, we look at $k$-generalized Fibonacci numbers which are perfect powers of exponent larger than $1$ of Lucas numbers. That is, we deal with the Diophantine equation \begin{equation*} F_{n}^{(k)}=L_{m}^{a} \end{equation*} in non-negative integers $k$, $n$, $m$, $a$, with $k\geq 3$, $m\geq 4$ and $a\geq 2$. We show that this equation has no solution under these conditions. The proof depends on lower bounds for linear forms in logarithms and some tools from Diophantine approximation.