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

Gaha Anouar; Mezroui Soufiane

paper

On the Non-negative Integer Solutions to Diophantine Equations $F_{n} - F_{m} = 7^{a}$ and $F_{n} - F_{m} = 1 3^{a}$

arXiv:2302.07262

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 $n > m ⩾ 0$ , $a ⩾ 2$ . 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.

On the Non-negative Integer Solutions to Diophantine Equations Fn​−Fm​=7a and Fn​−Fm​=13a

Abstract

On the Non-negative Integer Solutions to Diophantine Equations $F_{n} - F_{m} = 7^{a}$ and $F_{n} - F_{m} = 1 3^{a}$