The polynomials X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^{2} + (Y^3+Z^3)^2$ also capture their primes

Abstract

We show that there are infinitely many primes of the form X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^2+(Y^3+Z^3)^2$ . This extends the work of Friedlander and Iwaniec showing that there are infinitely many primes of the form X2+Y4$X^2+Y^4$ . More precisely, Friedlander and Iwaniec obtained an asymptotic formula for the number of primes of this form. For the sequences X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^2+(Y^3+Z^3)^2$ , 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 X2+(Y2+1)2$X^2+(Y^2+1)^2$ and X2+(Y3+Z3)2$X^2+(Y^3+Z^3)^2$ . Estimating the Type II sums is reduced to a counting problem that is solved by using the Weil bound, where the arithmetic input is quite different from the work of Friedlander and Iwaniec for X2+Y4$X^2+Y^4$ . We also show that there are infinitely many primes p=X2+Y2$p=X^2+Y^2$ where Y$Y$ is represented by an incomplete norm form of degree k$k$ with k−1$k-1$ variables. For this, we require a Deligne‐type bound for correlations of hyper‐Kloosterman sums.