Primes of the form $p^2 + nq^2$

TL;DR

This paper proves the infinitude of primes of the form p^2 + nq^2 for certain n, verifies a conjecture for n=4, and introduces new techniques involving Gowers norms and additive combinatorics.

Contribution

It establishes the infinitude of primes of specific quadratic forms and advances the method by applying recent Gowers norm developments to Type II sums.

Findings

Proved infinitely many primes of the form p^2 + nq^2 for n ≡ 0 or 4 mod 6.

Verified the Gaussian primes conjecture for n=4.

Developed new techniques using Gowers norms in the analysis of Type II sums.

Abstract

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec. We study the problem using the method of Type I/II sums in the number field $\mathbf{Q}(\sqrt{-n})$. The main innovation is in the treatment of the Type II sums, where we make heavy use of two recent developments in the theory of Gowers norms in additive combinatorics: quantitative versions of so-called concatenation theorems, due to Kuca and to Kuca--Kravitz-Leng, and the quasipolynomial inverse theorem of Leng, Sah and the second author.