Comparison of the lengths of the continued fractions of $\sqrt D$ and $\frac 12(1+\sqrt D)$

Abstract:

Let $D$ denote a positive nonsquare integer such that $D \equiv 1 (\bmod 4)$. Let $l(\sqrt D )$ (resp. $l(\tfrac {1} {2}(1 + \sqrt D ))$) denote the length of the period of the continued fraction expansion of $\sqrt D$ (resp. $\tfrac {1} {2}(1 + \sqrt D ))$). Recently Ishii, Kaplan, and Williams (On Eisenstein’s problem, Acta Arith. 54 (1990), 323-345) established inequalities between $l(\sqrt D )$ and $l(\tfrac {1} {2}(1 + \sqrt D ))$. In this note it is shown that these inequalities are best possible in a strong sense.