  In this note we present a construction of an infinite family of diagonal
quintic threefolds defined over $\Q$ each containing infinitely many rational
points. As an application, we prove that there are infinitely many quadruples
$B=(B_{0}, B_{1}, B_{2}, B_{3})$ of co-prime integers such that for a suitable
chosen integer $b$ (depending on $B$), the equation
$B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b$ has infinitely many
positive integer solutions.
