Imaginary quadratic fields of class number 2 and Levy-Hendy quadratics

Let d = pq equivalent to 3 (mod 4) with prime p, q and q < p. We prove a precise bound in Hendy's theorem on imaginary quadratic fields Q(root-d) with class number h(-d) = 2 which is a full analogue of Frobenius-Rabinowitsch theorem (the case of class number one). Namely it is shown that h( d) = 2 if and only if the Levy-Hendy quadratic L-d(x) = qx(2) qx+ l(0) with l(0) = p + q/4 takes only prime values for integers x in the interval 0 <= x <= l(0) - 2. We discuss also two related conjectures in connection with generalizing a result of Chowla et al. ([1]) on the class number and the least prime quadratic residue. (C) 2017 Elsevier Inc. All rights reserved.