A. Biró, On the class number one problem for special real quadratic number fields

Abstract. We consider certain families of real quadratic number fields with extremely small fundamental unit. For such families the class number one problem is analogous to the same problem for imaginary quadratic fields: Siegel`s theorem implies the finiteness of fields with class number one, but only ineffectively. We outline a method for the effective determination of the solutions, which leads, in particular, to the solution of a conjecture of Yokoi.