Class number one problem for a family of real quadratic fields

We effectively solve the class number one problem for a certain family Q(D)${\bf Q}(\sqrt D)$ (D is an element of F$(D\in {\cal F}$) of real quadratic fields, where F$ {\cal F}$ is an infinite subset of the set of odd positive fundamental discriminants. The set F${\cal F}$ contains the Yokoi discriminants n2+4$n2+4$, so our result is a generalization of the solution of Yokoi's Conjecture. But this family may contain also infinitely many fields with comparatively larger fundamental units than the fields in the Yokoi family (it may be as large as log2D$\log 2D$ instead of logD$\log D$). The proof is also a generalization of the proof of Yokoi's Conjecture