This work is licensed under the Creative Commons Attribution 4.0 International License.
Introduction and Preliminaries Section
Structure of
{\rm{\mathbb Q}}\left({\sqrt d} \right)
real quadratic number fields depend on the d > 0 positive non-square integer. It means that we have two different structures whether d ≡ 2,3(mod4) or d ≡ 1(mod4).
In this brief paper, we focus on certain types of
{\rm{\mathbb Q}}\left({\sqrt d} \right)
real quadratic fields for d ≡ 2,3(mod4) positive non-square integers. We define an integer sequence and determine such fields from parameterization of positive non-square integers d by using defined integer sequence. These types of real quadratic number fields contain the special written continued fraction expansion of the integral basis element wd such as
\left[ {{\gamma _0};\overline {11,\;11,\; \ldots,11,\;2{\gamma _0}}} \right]
, where period length is represented by l = l (d) of wd.
We also demonstrate fundamental units of such fields in the case of d ≡ 2,3(mod4). All the results help us to get a practical and handy method to calculate continued fraction expansions and fundamental units.
We refer all references [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] to readers for more information on the structure of quadratic fields.
In this section, we give following basic notations to use in our Main Results section.
Definition 1.1
Let {ℵi} be an integer sequence. It is defined by the recurrence relation
{\aleph _i} = 11{\aleph _{i - 1}} + {\aleph _{i - 2}}
with the seed values ℵ0 = 0 and ℵ1 = 1 for i ≥ 2.
Lemma 1.1
Let d be a square-free positive integer such that d ≡ 2, 3(mod4). If we put
{w_d} = \sqrt d
and
{\gamma _0} = \sqrt d
into wR = γ0 + wd, then we get wd ∉ R(d) but wR ∈ R (d).
Furthermore, for the period l = l (d) of wR, we have continued fraction expansion of
{w_R} = \left[ {\overline {\;2{\gamma _0},{\gamma _1},{\gamma _2}, \ldots,\;{\gamma _{l\left(d \right) - 1}}}} \right]
as pur-periodic and continued fraction expansion of wd as periodic
{w_d} = \left[ {{\gamma _0};\overline {\;{\gamma _1},{\gamma _2}, \ldots,\;{\gamma _{l\left(d \right) - 1}},\;2{\gamma _0}}} \right]
. Besides, let
{w_R} = {{{w_R}{P_l} + {P_{l - 1}}} \over {{w_R}{Q_l} + {Q_{l - 1}}}} = \left[ {\;\;2{\gamma _0},{\gamma _1},{\gamma _2}, \ldots,\;{\gamma _{l\left(d \right) - 1}},{w_R}} \right]
be a modular automorphism of wR. Then, the fundamental unit ɛd of
\left({\sqrt d} \right)
real quadratic number field is given by the following formula:
\matrix{{{\varepsilon _d} = {{{t_d} + {u_d}\sqrt d} \over 2} = \;\left({{\gamma _0} + \sqrt d} \right){Q_{l\left(d \right)}} + {Q_{l\left(d \right) - 1}}} \cr {{t_d} = 2{\gamma _0}{Q_{l\left(d \right)}} + 2{Q_{l\left(d \right) - 1}}\,\,{\rm{and}}\,\,{u_d} = 2{Q_{l\left(d \right)}}}}
where Qi is determined by Q0 = 0, Q1 = 1 and Qi+1 = γiQi + Qi−1 for i ≥ 1.
Note.I (d) is a set of all quadratic irrational numbers in
\left({\sqrt d} \right)
. α in I (d) is reduced if α > 1 and −1 < α′ < 0 (α′ is the conjugate of α). R(d) is also a set of all reduced quadratic irrational numbers in I (d). Besides, for any number α, R(d) is purely periodic in the continued fraction expansion.
Main Results
In this section, we give two theorems and two corollaries, which carry out the main aim of the brief paper.
Theorem 2.1
Let d be a square-free positive integer and 𝓁 ≥ 2 be a positive integer such that it is not divided by three. Supposing that the parameterization of d is
d = {\left({{{11 + \left({2J + 1} \right){\aleph _\ell}} \over 2}} \right)^2} + \left({2J + 1} \right){\aleph _{\ell - 1}} + 1
where J ≥ 0 is a positive integer.
If 𝓁 ≡ 2, 4, 5 (mod6) and J ≥ 0 is a even positive integer, then d ≡ 2, 3 (mod4). Besides, we obtain
{w_d} = \left[ {{{\left({2J + 1} \right){\aleph _\ell} + 11} \over 2};\,\overline {\mathop {\underbrace {11,11, \ldots,11}_{\ell - 1}},\left({2J + 1} \right){\aleph _\ell} + 11}} \right]
with 𝓁 = 𝓁(d) for d ≡ 2, 3(mod4). Furthermore, we have the fundamental unit ɛd as follows:
{\varepsilon _d} = \left({{{11 + \left({2J + 1} \right){\aleph _\ell}} \over 2}{\aleph _\ell} + {\aleph _{\ell - 1}}} \right) + {\aleph _\ell}\sqrt d.
Theorem 2.2
Assume that d is a square-free positive integer, period length is 𝓁 ≥ 2 and 3 ∤𝓁. The parameterization of d is given by
d = {\left({{{11 + \left({2J + 1} \right){\aleph _\ell}} \over 2}} \right)^2} + \left({2J + 1} \right){\aleph _{\ell - 1}} + 1
where J ≥ 0 is a positive integer. If 𝓁 ≡ 1 (mod6) and J is odd positive integer, then d ≡ 2(mod4). Also, we have same form of the wd defined in Theorem 2.1, while 𝓁 = 𝓁(d) and d ≡ 2(mod4). In addition, we obtain the coefficients of fundamental unit td and ud as follows:
{t_d} = \left({2 + 1} \right)\aleph _\ell ^2 + 11{\aleph _\ell} + 2{\aleph _{\ell - 1}}\,\,{\rm{and}}\,\,{u_d} = 2{\aleph _\ell}.
Corollary 2.1
Let d be square-free positive integer and 𝓁 ≥ 2 be a positive integer satisfying the conditions of Theorem 2.1. Let the parameterization of d be defined as
d = {\left({{{11 + {\aleph _\ell}} \over 2}} \right)^2} + {\aleph _{\ell - 1}} + 1
. Then, we obtain d ≡ 2, 3(mod4), and the continued fraction expansion of wd is given by
{w_d} = \left[ {{{11 + {\aleph _\ell}} \over 2};\overline {\mathop {\underbrace {11,\;11,\; \ldots,\;11}_{\ell - 1}},11 + {\aleph _\ell}}} \right]
for 𝓁 = 𝓁(d). Also, we get the fundamental unit as
{\varepsilon _d} = \left({{{11 + {\aleph _\ell}} \over 2} + \sqrt d} \right){\aleph _\ell} + {\aleph _{\ell - 1}}
.
Proof
It is obtained by Theorem 2.1, if we chose J = 0. By the way, we prepare the following table, which includes some of infinite numerical examples considering condition 𝓁 ≡ 2, 4, 5 (mod6) for Corollary 2.1.
Let d be a square-free positive integer and 𝓁 > 1 and 𝓁 ≡ 5(mod6). Assume that the parameterization of d is given by
d = {\left({{{11 + 3{\aleph _\ell}} \over 2}} \right)^2} + 3{\aleph _{\ell - 1}} + 1
Then, we obtain d ≡ 2(mod4) and
{w_d} = \left[ {{{11 + 3{\aleph _\ell}} \over 2};\overline {\mathop {\underbrace {11,11, \ldots,11}_{\ell - 1}},11 + 3{\aleph _\ell}}} \right]
for 𝓁 = 𝓁(d). Also, we obtain following equation for the coefficient of fundamental units td and ud.
{t_d} = 3\aleph _\ell ^2 + 11{\aleph _\ell} + 2{\aleph _{\ell - 1}}\,\,{\rm{and}}\,\,{u_d} = 2{\aleph _\ell}
Proof
We get this corollary from Theorem 2.2 if J is chosen as J = 1. Besides, we prepare the following table, which contains several of infinite numeric illustrations under the conditions of 𝓁 > 1 and 𝓁 ≡ 1 (mod6) for Corollary 2.2. We cannot write 𝓁 = 7 in the table since d = 763180879250 has a square factor.
Remark
Readers can see that the real quadratic fields depend on two different parameters such as period length 𝓁 = 𝓁(d) and J ≥ 0 integer. So, we can determine infinitely many real quadratic number fields with their structures if we change the values of these parameters.
In the topic of real quadratic fields, there are some tools such as fundamental unit and continued fraction expansion That are useful for determining structures of such fields. The main aim of this paper was to provide a practical method to calculate fundamental unit rapidly and simply for such real quadratic number fields. We are sure that this paper will be useful for readers.
