Nonnegative integer solutions of the equation $F_n-F_m=5^a$

Abstract

In this study, we solve the Diophantine equation in the title in nonnegative integers m; n; and a. Thesolutions are given by $F_{1-}F_0=F_2-F_0=F_3-F_2=F_3-F_1=F_4-F_3=5^0$ and $F_{5-}F_0=F_6-F_4=F_7-F_6=5$ Then we give a conjecture that says that if $ageq2$ and $p>7$ is prime, then the equation $F_n-F_m=p^a$ has no solutionsin nonnegative integers m, n.