Some results on a special type of real quadratic fields

Abstract

In this paper, we determine the real quadratic fields Q(√d ) coincide with positive square - free integers d including the continued fraction expansion form of w_d = [a₀ ; 7,7,…,7 l-1 , a_l]. Furthermore, we deal with determining fundamental units and Yokoi’s d -invariants n_d and m_d in the relation to continued fraction expansion of w_d where l( d ) is a period length of w_d for the such type of real quadratic number fields Q(√d ) . The present paper improve the theory of fundamental unit which generates the unit group of real quadratic fields and also determine the special form of continued fraction expansion of integral basis element in real quadratic fields.