In this note we present a construction of an infinite family of diagonal quintic threefolds defined over
Q
\mathbb {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
)
B=(B_{0}, B_{1}, B_{2}, B_{3})
of co-prime integers such that for a suitable chosen integer
b
b
(depending on
B
B
), the equation
B
0
X
0
5
+
B
1
X
1
5
+
B
2
X
2
5
+
B
3
X
3
5
=
b
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.