On Eisenstein primes

TL;DR

This paper proves the infinitude of primes of a specific quadratic form with an additional prime condition, extending classical results by employing algebraic number theory techniques in Eisenstein integers.

Contribution

It introduces a novel approach using Eisenstein integers to establish the infinitude of primes of a quadratic form with a prime linear combination, generalizing previous methods.

Findings

Proves infinitely many primes of the form ll^2 - ll m + m^2 with 2ll - m prime

Adapts techniques from Fouvry and Iwaniec to Eisenstein integers

Extends classical prime distribution results to new quadratic forms

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.