On prime values of binary quadratic forms with a thin variable

Abstract

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