There are infinitely many primes of the form $x^2+y^2+1$, as proved by Bredihin. Motohashi improved the result by showing that there were $\gg x/\log^2 x$ such primes up to $x$. But we expect $\Theta(x/\log^{3/2}x)$ primes up to $x$; has this result been proved? (Failing that, is anything else known about the density since the 70s? I know Sun & Pan proved the Green-Tao theorem for these primes.)



Primes of the form $x^2 + y^2 + 1$
[Iwaniec](http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21118.pdf) (Acta Arith. 1972) showed that the number of such primes has the order $x/(\log x)^{3/2}$. His result applies more generally to translates of binary quadratic forms. One can also show that short intervals contain the right density of such primes: for example, see the work of [Matomaki](http://users.utu.fi/ksmato/papers/Primesm2n2p1.pdf) (Acta Arith. 2007). 


