A number of solutions of the equation, which contains quadratic forms with a growing discriminant

In this article, an analogue of the Ingam binary additive divisor problem is considered. A binaryadditive problem with quadratic forms is studied. The asymptotical formula of the number of solution of diophantine equation $Q_1(\bar{m})-Q_2(\bar{k})=1$ is received. This equation contains binarypositive defined primitive qua√dratic forms $Q_1(\bar{m})$ and $Q_2(\bar{k})$ corresponded to the ideal class ofimaginary quadratic fields $Q(\sqrt{d}). The discriminant of an imaginary quadratic field is a growing-parameter. The number of solutions searched with weights $\exp(-(Q_1(\bar{m})+Q_2(\bar{k}))/n)$ with the growth of the parameter n. Proof of the asymptotical formula is carried out by the circular method. The estimation of Gauss double sums with growing discriminant and estimation of Kloosterman's sum by A. Weil are used.