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 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 B0Xo5 + B1X15 + B2X25 + B3X35 = b has infinitely many positive integer solutions.