  We show that there are infinitely many primes of the form $X^2+(Y^2+1)^2$ and
$X^2+(Y^3+Z^3)^2$. This extends the work of Friedlander and Iwaniec showing
that there are infinitely many primes of the form $X^2+Y^4$. More precisely,
Friedlander and Iwaniec obtained an asymptotic formula for the number of primes
of this form. For the sequences $X^2+(Y^2+1)^2$ and $X^2+(Y^3+Z^3)^2$ we
establish Type II information that is too narrow for an aysmptotic formula, but
we can use Harman's sieve method to produce a lower bound of the correct order
of magnitude for primes of form $X^2+(Y^2+1)^2$ and $X^2+(Y^3+Z^3)^2$.
Estimating the Type II sums is reduced to a counting problem which is solved by
using the Weil bound, where the arithmetic input is quite different from the
work of Friedlander and Iwaniec for $X^2+Y^4$. We also show that there are
infinitely many primes $p=X^2+Y^2$ where $Y$ is represented by an incomplete
norm form of degree $k$ with $k-1$ variables. For this we require a
Deligne-type bound for correlations of hyper-Kloosterman sums.
