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=n ²+4 is a square-free positive integer with n=m(m ²-306) for some odd m, with the extra condition (dN)=-1 for N=2 ³{dot operator}3 ³{dot operator}103{dot operator}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=n ²+4 could be interesting having in mind that even the class number two problem for these discriminants is not yet solved unconditionally.