The polynomials $X^2+(Y^2+1)^2$ and $X^{2} + (Y^3+Z^3)^2$ also capture their primes

Abstract: 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.