Quadratic irrationals with fixed period length in the continued fraction expansion

Abstract

We present an algorithm that makes it possible to write out all quadratic irrationals of the form\(\sqrt D \), that have a given even period length in the continued fraction expansion. It turns out that in the expansion $$\sqrt D = \left[ {b_0 ,\overline {l_1 ,...,l_L ,...,l_1 ,2b_0 } } \right]$$ λ={l1, ..., lL+1} is almost arbitrary, and b0 (and, consequently D) runs through a very narrow sequence depending on λ. We obtain a summation formula for the class numbers of indefinite binary forms with discriminant D with D≤X for which the set λ is fixed.