D. R. Heath-Brown and B. Z. Moroz

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.

2000 Mathematical Subject Classification: 11N32 (primary), 11N36, 11R44 (secondary).

Keywords: primes, binary cubic polynomials, cubic fields.