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

TL;DR

This paper proves the infinitude of primes of specific polynomial forms, extending previous work, and introduces new bounds and methods involving Harman's sieve, Weil bounds, and hyper-Kloosterman sums.

Contribution

It extends the class of polynomial forms known to generate infinitely many primes and develops novel bounds and techniques for counting such primes.

Findings

Infinitely many primes of the form X^2+(Y^2+1)^2 and X^2+(Y^3+Z^3)^2 are proven.

Established lower bounds for the number of such primes using Harman's sieve.

Derived bounds for correlations of hyper-Kloosterman sums related to primes of the form X^2+Y^2.

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.