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

Dedicated to the occasion of John Friedlander's 80th birthdayWe 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 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.MSC2020: primary 11N32