Prime values of f(a,b2) and f(a,p2), f quadratic

Xiao, Stanley Yao

doi:10.2140/ant.2024.18.1619

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Prime values of $f(a,b^2)$ and $f(a,p^2)$, $f$ quadratic

Stanley Yao Xiao

Vol. 18 (2024), No. 9, 1619–1679

DOI: 10.2140/ant.2024.18.1619

Abstract

We prove an asymptotic formula for primes of the shape $f (a, b^{2})$ with $a$ , $b$ integers and of the shape $f (a, p^{2})$ with $p$ prime. Here $f$ is a binary quadratic form with integer coefficients, irreducible over $ℚ$ and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form $x^{2} + y^{4}$ and of Heath-Brown and Li on primes of the form $a^{2} + p^{4}$ , as well as earlier work of the author with Lam and Schindler on primes of the form $f (a, p)$ with $f$ a positive definite form.

Keywords

primes, prime values of quadratic forms, Friedlander–Iwaniec theorem

Mathematical Subject Classification

Primary: 11N32

Secondary: 11N35, 11N36, 11R45

Milestones

Received: 17 July 2022

Revised: 30 August 2023

Accepted: 12 October 2023

Published: 19 September 2024

Authors

Stanley Yao Xiao
Department of Mathematics and Statistics University of Northern British Columbia Prince George Canada

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Vol. 18, No. 9, 2024