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.
The proof makes 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 speaker.