Explicit formulas for units in certain quadratic number fields

Abstract

There is a class of quadratic number fields for which it is possible to find an explicit continued fraction expansion of a generator and hence an explicit formula for the fundamental unit. One therewith displays a family of quadratic fields with relatively large regulator. The formula for the fundamental unit seems far simpler than the continued fraction expansion, yet the expansion seems necessary to show the unit is fundamental. I explain what is going on and go some way towards taming the sequence of ever more complicated arguments of Yamomoto, Hendy, Bernstein, Williams, Levesque and Rhin, Levesque, Azuhata, Halter-Koch, Mollin and Williams, and Williams.