An exponential Diophantine equation related to the difference of powers of two Fibonacci numbers

TL;DR

This paper proves that for any x >= 4, the difference of the x-th powers of two consecutive Fibonacci numbers cannot be a Lucas number, establishing a specific non-existence result in number theory.

Contribution

The paper introduces a new non-existence theorem linking powers of Fibonacci numbers and Lucas numbers for x >= 4.

Findings

No solutions for x >= 4 where the difference of x-th powers of consecutive Fibonacci numbers equals a Lucas number.

Establishes a specific exponential Diophantine equation has no solutions beyond a certain bound.

Contributes to the understanding of relationships between Fibonacci and Lucas numbers in exponential equations.

Abstract

In this paper, we prove that there is no x>=4 such that the difference of x-th powers of two consecutive Fibonacci numbers greater than 0 is a Lucas number.