The binary Goldbach problem with one prime of the form p=k2+l2+1

We prove that almost all integers n≡0 or 4 (mod 6) can be written in the form n=p1+p2, where p1=k2+l2+1 with (k,l)=1. The proof is an application of the half-dimensional and linear sieves with arithmetic information coming from the circle method and the Bombieri–Vinogradov prime number theorem.