On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues 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 cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).