A note on the fundamental unit in some types of the real quadratic number fields

Abstract

Let k=Q(d) be a real quadratic numbefield where d > 0 is a positive square-free integer. The map d→Q(d) is a bijection from the set off all square-free integers d ≠ 0, 1 to the set of all quadratic fields Q(d)={x+yd|x,y∈Q}. Furthermore, integral basis element of algebraic integer’s ring in real quadratic fields is determined by either wd=d=[a0;a1,a2,⋯,al(d)−1,2a0¯] in the case of d ≡ 2,3(mod 4) or wd=1+d2=[a0;a1,a2,⋯,al(d)−1,2a0−1¯] in the case of d ≡ 1(mod 4) where l(d) is the period length of continued fraction expansion.The purpose of this paper is to obtain classification of some types of real quadratic fields Q(d), which include the specific form of continued fraction expansion of integral basis element wd, for which has all partial quotient elements are equal to each other and written as ξs (except the last digit of the period) for ξ positive even integer where period length is l=l(d) and d ≡ 2,3(mod 4) is a square free positive integer. Moreover, the present paper deals with determining new certain...