Primes Represented by Binary Cubic Forms

Let f(x, y) be a binary cubic form with integral rational coefficients,and suppose that the polynomial f(x, y) is irreducible in Qx,y] and no prime divides all the coefficients of f. We provethat the set f Z⁽²⁾ contains infinitely many primes unless f(a,b) is even for each (a,b) in Z², in which case the set contains infinitely many primes. 2000Mathematical Subject Classification: primary 11N32; secondary11N36, 11R44.