On prime values of binary quadratic forms with a thin variable

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 $\mathrm{\ell },m\in \mathbb{Z}$ such that both $F(\mathrm{\ell },m)$ and $G(\mathrm{\ell },m)$ are primes as long as there are no local obstructions.