In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002) 257–288) we showed that an irreducible integral binary cubic form $f(x, y)$ attains infinitely many prime values, providing that it has no fixed prime divisor. We now extend this result by showing that $f(m, n)$ still attains infinitely many prime values if $m$ and $n$ are restricted by arbitrary congruence conditions, providing that there is still no fixed prime divisor.
Two immediate consequences for the solvability of diagonal cubic Diophantine equations are given.
