Comparison of the lengths of the continued fractions of √𝐷 and \frac12(1+√𝐷)

Abstract

Let D D denote a positive nonsquare integer such that D ≡ 1 ( mod 4 ) D \equiv 1\,(\bmod 4) . Let l ( D ) l(\sqrt D ) (resp. l ( 1 2 ( 1 + D ) ) l(\tfrac {1} {2}(1 + \sqrt D )) ) denote the length of the period of the continued fraction expansion of D \sqrt D (resp. 1 2 ( 1 + D ) ) \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 ( D ) l(\sqrt D ) and l ( 1 2 ( 1 + D ) ) l(\tfrac {1} {2}(1 + \sqrt D )) . In this note it is shown that these inequalities are best possible in a strong sense.