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

2023/02/14 by Anouar, Gaha, Soufiane, Mezroui
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/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\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.

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