The Gauss class number problem is to determine a complete list of quadratic number fields for any given class number. It follows from Siegel`s theorem that for each class number there are only finitely many imaginary quadratic fields (IQFs) and real quadratic fields of Richaud-Degert type (RQFs of R-D type). Since Siegel`s theorem is ineffective, it cannot provide a solution for the Gauss class number problem. Goldfeld discovered an effective method, which concerns arithmetic of an elliptic curve, to solve the class number problem for IQFs and RQFs of R-D type. For IQFs only, Oesterlé simplified Goldfeld`s proof and made an explicit result, which led him to solve the class number three problem for IQFs. We find explicit constants in Goldfeld`s method and apply the results to the class number problem for RQFs of R-D type.