Primes represented by binary cubic forms (Q2759197)

Let \(f(x,y)\) be a binary cubic form with integral rational coefficients, and suppose that the polynomial \(f(x,y)\) is irreducible in \(\mathbb Q[x,y ]\) and no prime divides all the coefficients of \(f\). We prove that the set \(f (\mathbb Z^2)\) contains infinitely many primes unless \(f(a,b)\) is even for each \((a,b)\) in \(\mathbb Z^2\), in which case the set \(\frac{1}{2} f(\mathbb Z^2)\) contains infinitely many primes. This theorem is a generalization of the recent theorem of \textit{D. R. Heath-Brown} [Acta Math. 186, 1--84 (2001; Zbl 1007.11055)] on the infinitude of the primes represented by the cubic form \(x^3+ 2y^3\), and its proof follows the pattern set up in that paper. The main innovations are related to the arithmetic of a general cubic field; our considerations require, in particular, some techniques from E. Hecke's multidimensional arithmetic which we briefly review. As in the cited work of Heath-Brown, we have actually obtained an asymptotic formula for the relevant number of primes.