The polynomials X2+(Y2+1)2 and X2 + (Y3+Z3)2 also capture their primes
Abstract
We show that there are infinitely many primes of the form X2+(Y2+1)2 and X2+(Y3+Z3)2. This extends the work of Friedlander and Iwaniec showing that there are infinitely many primes of the form X2+Y4. More precisely, Friedlander and Iwaniec obtained an asymptotic formula for the number of primes of this form. For the sequences X2+(Y2+1)2 and X2+(Y3+Z3)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 and X2+(Y3+Z3)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 X2+Y4. We also show that there are infinitely many primes p=X2+Y2 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.
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