On the period length of the continued fraction expansion of a quadratic irrational and the class number of real quadratic fields

The fundamental result of the paper is the following theorem: suppose that the Riemann conjecture is valid for the Dedekind $\xi$-functions of all fields $\mathbb{Q}\Bigl(\Bigl(\frac{1+\sqrt{5}}{2}\Bigr)^{1/k},1^{1/k}\Bigr)$ Then there exists a constant $C>0$ such that on the interval $p\leq x$ one can find at least $Cx\log^{-1}x$ prime numbers $p$ for which $h(Sp^2)=2$. Here $h(d)$ is the number of proper equivalence classes of primitive binary quadratic forms of discriminant $d$. In addition, it is proved that
$$ \sum_{p\leq x}h(Sp^2)\log p=O(x^{3/2}). $$
For these sequence of discriminants of a special form with increasing square-free part, one has obtained a nontrivial estimate from above for the number of classes.