  In this note we will analyze a diophantine equation raised by Michael Bennett
in [1] that is pivotal in establishing that powers of five has few digits in
its ternary expansion. We will show that the Diophantine equation
$3^{a}+3^{b}+2=n^5$, where $(n,3)=1$ and $a>b>0$ is insoluble for pairs of
positive integers $(a,b)$ where they are both even or one is even and the other
is odd. In the case where both $(a,b)$ are odd, there is one known solution
$2^5=3^3+3^1+2.$ We will show that there are no other solutions to the
diophantine equation for $n^{5}<32\left(1+3(10^6)\right)^5$.
