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

TL;DR

This paper completely solves a nonlinear Diophantine equation involving powers of three consecutive $k$-Lucas numbers, providing a full characterization of solutions for all fixed $k extgreater 1$ and nonnegative integers.

Contribution

It offers the first complete solution to a nonlinear Diophantine equation involving powers of three consecutive $k$-Lucas numbers for all fixed $k extgreater 1$.

Findings

Explicit solutions characterized for all fixed $k extgreater 1$.

The equation has finitely many solutions in nonnegative integers.

Methodology can be applied to similar nonlinear equations involving generalized 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 $k\ge 2$.