On Eisenstein primes

Abstract

In this paper, we prove that there are infinitely many primes of the form ${\ell }^2-\ell m+m^2$ such that $2\ell -m$ is also prime. To prove this, we follow along the lines of the work of Fouvry and Iwaniec (1997) who showed that there are infinitely many primes of the form ${\ell }^2+m^2$ for $\ell$ prime, but use $\mathbb{Z}[\omega ]$ instead of $\mathbb{Z}[i]$ to work with the bilinear forms that arise in both.