On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers

Abstract: Let $(L_n^{{(k)})_{n\geq} 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we completely solve the nonlinear Diophantine equation $\left(L_{n+1}^{{(k)}\right)x+\left(L_{n}{(k)}\right)x-\left(L_{n-1}{(k)}\right)x=L_m{(k)}$,} in nonnegative integers $n$, $m$, $k$, $x$, with $k\ge 2$.