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

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.

Automated summary

This paper proves the existence of infinitely many prime numbers 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, who showed the existence of infinitely many prime numbers of the form X^2+Y^4 and obtained an asymptotic formula for them.