Primes Represented by Binary Cubic Forms

Abstract

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