**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.