This paper establishes the result that numbers of the shape 5(2)p(p = 2 mod
9) and 5(2)p(2)(p = 5 mod 9), where p a prime, are expressible as the sum
of two rational cubes by finding modular parametrizations of the correspond
ing elliptic curves, picking special complex points on the curve and manipu
lating them to produce rational points thereon.