On the Exponential Diophantine Equation $(F_{m+1}^{(k)})^x-(F_{m-1}^{(k)})^x = F_n^{(k)}$

TL;DR

This paper explicitly solves a specific exponential Diophantine equation involving generalized Fibonacci numbers, employing advanced number theory techniques such as bounds on linear forms in logarithms and continued fractions.

Contribution

It provides a complete solution to the equation using modern Diophantine approximation methods, extending previous research in the area.

Findings

All solutions to the equation are explicitly determined.

The methods combine bounds on linear forms in logarithms with continued fraction properties.

The approach improves upon previous partial results.

Abstract

In this paper, we explicitly find all solutions of the title Diophantine equation, using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker-Davenport reduction method in Diophantine approximation, due to Dujella and Peth\"o. This paper extends the previous work of \cite{Patel}.