  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.
