Construction of diagonal quintic threefolds with infinitely many rational points

In this note we present a construction of an infinite family of diagonal quintic threefolds defined over $\text{Q}$ 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.