The main purpose of this article is to present a numerical data which shows relations between real quadratic fields of class number 1 and a mysterious behavior of the period of simple continued fraction expansion of certain quadratic irrationals. For that purpose, we define a class number, a fundamental unit,a discriminant and a Yokoi invariant for a non-square positive integer, and then see that a generalization of theorems of Siegel and of Yokoi holds. These and a theorem of Friesen and Halter-Koch imply several interesting conjectures for solving Gauss' class number problem for real quadratic fields.
Fuminori KAWAMOTO. Koshi TOMITA. "Continued Fractions and Gauss' Class Number Problem for Real Quadratic Fields." Tokyo J. Math. 35 (1) 213 - 239, June 2012. https://doi.org/10.3836/tjm/1342701351