  We show under the assumption that the Tate-Shafarevich group of any elliptic
curve over the rational numbers is finite that the cubic surface $x_1^3 +
p_1p_2x_2^3 + p_2p_3x_3^3 + p_3p_1x_4^3 = 0$ has a rational point, where $p_1,
p_2$ and $p_3$ are rational primes congruent to $2$ or $5$ modulo $9$.

Number Theory