Effective lower bound for the class number of a certain family of real quadratic fields

In this work we establish an effective lower bound for the class number of the family of real quadratic fields Q(d), where d=n2+4 is a square-free positive integer with n=m(m2−306) for some odd m, with the extra condition (dN)=−1 for N=23⋅33⋅103⋅10303. This result can be regarded as a corollary of a theorem of Goldfeld and some calculations involving elliptic curves and local heights. The lower bound tending to infinity for a subfamily of the real quadratic fields with discriminant d=n2+4 could be interesting having in mind that even the class number two problem for these discriminants is not yet solved unconditionally.