On prime values of binary quadratic forms with a thin variable

TL;DR

This paper extends previous work on prime values of the form x^2 + y^2 to general primitive positive definite binary quadratic forms, showing infinitely many primes occur under certain conditions.

Contribution

It generalizes the result of Fouvry and Iwaniec to arbitrary primitive positive definite binary quadratic forms, broadening the scope of prime value results.

Findings

Infinitely many primes represented by general quadratic forms and linear forms.

Existence of primes under no local obstructions.

Extension of classical results to a wider class of forms.

Abstract

In this paper we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form $x^2 + y^2$ with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form $F$ and binary linear form $G$, there exist infinitely many $\ell, m\in\mathbb{Z}$ such that both $F(\ell, m)$ and $G(\ell, m)$ are primes as long as there are no local obstructions.