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 x2+y2 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 ℓ,m∈Z such that both F(ℓ,m) and G(ℓ,m) are primes as long as there are no local obstructions.