On simultaneous representations of primes by binary quadratic forms

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M² + 7N², and knowing M and N makes it possible to “predict” whether p = A² + 14B² is solvable or p = 7C² + 2D² is solvable. More generally, let q and r be distinct primes, and let an integral solution of H²p = M² + qN² be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K²q = A² + qrB² is solvable and the possible values of K′ for which K′²p = qC² + rD² is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant ?4qr from representing p.