On the Non-negative Integer Solutions to Diophantine Equations $F_n - F_m = 7^a$ and $F_n - F_m = 13^a$

TL;DR

This paper proves that the Diophantine equations involving differences of Fibonacci numbers equaling powers of 7 or 13 have no solutions, confirming a previous conjecture using advanced number theory techniques.

Contribution

The paper confirms a conjecture by proving the non-existence of solutions for specific Fibonacci difference equations involving powers of 7 and 13.

Findings

No solutions exist for the equations $F_n - F_m = 7^a$ and $F_n - F_m = 13^a$ with the given constraints.

Utilizes Baker's theory and Diophantine approximation methods to establish non-existence.

Provides a rigorous proof confirming the conjecture of Erduvan and Keskin.

Abstract

In this paper, we study the solutions of the equation $F_n-F_m=p^a$ where $p$ is either $7$ or $13$ and $n>m\geqslant 0$, $a\geqslant 2$. We confirm the conjecture of Erduvan and Keskin by proving that there is no solutions for this Diophantine equation. We will use the lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in Diophantine approximation.