The new approximations on the class number problem of real quadratic number fields by yokoi's invariant values

The aim of this study is to investigate the class numbers of real quadratic number fields depending on Yokoi's invariant values defined by the coefficients of the fundamental units of such fields. First of all, an important criterion is determined for the first term of the symmetric part on the continued fraction expansion of $w_d$, integral basis element of $\mathbb{Q}(\sqrt{d})$ real quadratic number field where $d \equiv 1 (mod 4)$ positive square-free integer. Thus, some lower and upper bounds for coefficients of the fundamental units of such fields are given depending on Yokoi's invariant values. The bounds obtained for the fundamental unit are applied to Dirichlet Class Number Formula and some new criteria on the class number of the field are obtained. Therefore, new properties regarding the algebraic structures of these fields have been obtained while bringing the new approaches to the class numbers of real quadratic number fields. Finally, it has been shown that Ankeny-Artin-Chowla Conjecture is true for some cases expressed in terms of Yokoi's invariant values.