SOME RESULTS ON SPECIAL CONTINUED FRACTION EXPANSIONS IN REAL QUADRATIC NUMBER FIELDS

Abstract

The aim of this paper is to determine and investigate the continued fractions expansions of wd for the real quadratic number fields Q(root d) for which the period has constant elements that are completely equal to 2 (except the last digit of period) in the symetric part of the period of integral basis element where d 2, 3 mod 4 is a square free positive integer. Moreover, we give new explicit formulas for the fundamental unit ed and Yokoi's d-invariants rid and and in relation to continued fraction expansion of such form of w(d). These new formulas are not known in the literature of real quadratic fields. Such types of real quadratic fields are classified as new results.