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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 theconjecture of Erduvan and Keskin by proving that there is no solutions for thisDiophantine equation. We will use the lower bounds for linear forms inlogarithms (Baker's theory) and a version of the Baker-Davenport reductionmethod in Diophantine approximation.

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