Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields

Abstract

In 2002, Tomita and Yamamuro defined several theorems for fundamental unit of certain real quadratic number fields. Although, there are infinitely many values of d having all 1s in the symmetric part of continued fraction expansion of w(d), Tomita and Yamamuro (1992) had described explicitly one type of d for the fundamental units of the real quadratic fields by using Fibonacci sequence in the Theorem 3 for d equivalent to 2,3(mod4) and in the Theorem 2 in the case of d equivalent to 1(mod4) (2002). The main purpose of this paper is to generalize and provide an improvement of the theorem 3 and the theorem 2 in the paper of Tomita and Yamamuro (2002). Moreover, the present paper deals with new certain formulas for fundamental unit epsilon(d) and Yokoi's d-invariants n(d), m(d) in the relation to continued fraction expansion of w(d) for such real quadratic fields. All results are supported by numerical tables.