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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
Abstract

We prove an asymptotic formula for primes of the shape f(a,b2) with a, b integers and of the shape f(a,p2) with p prime. Here f is a binary quadratic form with integer coefficients, irreducible over and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form x2 + y4 and of Heath-Brown and Li on primes of the form a2 + p4, 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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