Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields
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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.