Abstract

In this paper, we consider the certain types of real quadratic fields where is a square free positive integer. We obtain new parametric representation of the fundamental unit for such types of fields. Also, we get a fix on Yokoi’s invariants as well as class numbers and support all results with tables.

1. Introduction

Quadratic fields have many applications to different fields of mathematics which contain algebraic number theory, algebraic geometry, algebra, cryptology, and also other scientific fields like computer science. It is also well known that the fundamental units play an important role in studying the class number problem, unit group, pell equations, cryptology, network security and even computer science.

Recently, in ¹, Benamar and his co-authors worked on a type of special monic and non square free polynomials related with fixed period continued fraction expansion of square root of rational integers. In ², Clemens with collaborators proved explicit continued fractions with almost periodic or almost symmetric patterns in their partial quotients, and infinite series whose terms satisfy certain recurrence relations using Newton′s method. Tomita and Kawamato ⁵ constructed an infinite family of real quadratic fields with large even period of minimal type with class number. Zhang and Yue ²² investigated fundamental unit with positive norm as well as several congruence relations about the coefficient of fundamental unit. Halter-Koch ⁴ studied on a construction of infinite families of real quadratic fields with large fundamental units. Yokoi defined several invariants which two of them were significant invariants and determined as and by using the coefficients of fundamental unit ^{18, 19, 20, 21}. The author ^{10, 11}, has obtained several important results on fundamental units and Yokoi’s invariants for special type of in the case of

We also refer other significant references which were worked on the fundamental unit, prime producing polynomials, class numbers problem, continued fraction expansions etc… to the readers for more information and background about the quadratic fields.

The fundamental unit of the ring of algebraic integers in a real quadratic number field is a generator of the group of units. Furthermore, integral basis element of algebraic integer’s ring in real quadratic fields is determined by either

in the case of or

in the case of , where is the period length of continued fraction expansion.

For the set of all quadratic irrational numbers in , we say that in is reduced if , ( is the conjugate of with respect to ), and denote by the set of all reduced quadratic irrational numbers in . Then, it is well known that any number in is purely periodic in the continued fraction expansion and the denominator of its modular automorphism is equal to fundamental unit of . Also, Yokoi’s invariants, which were defined by H.Yokoi are determined by the coefficient of fundamental unit as and where represents the greatest integer not greater than (floor of ).

Continued fraction expansion of integral basis elemet has got two different forms accordıng to is a square free positive integer congruent to 2,3 modulo 4 or to congruent 1 modulo 4. Besides, the partial constants of continued fraction expansions create the different forms of for real quadratic number fields.

The aim of this paper is to classify some types of real quadratic number fields where is a positive square free integer. Such real quadratic fields include the continued fraction expansion of the integral basis element which has got partial constant elements are same and written as nines (except the last digit of the period). The representation of fundamental unit is determined for such types of real quadratic fields using the parametrization of positive square free integers (not only in the case of but also in the case of ). Also, the present paper deals with computing Yokoi’s invariants and as well as class numbers. Using the practical way, the results obtained in this paper are supported by numerical tables.

2. Prelimineries

In this section we also give some fundamental concepts for the proof of our main theorems defined in the next section.

Definition 2.1. is called as a sequence defined by the recurrence relation

with the initial conditions and for

Lemma 2.2. Let be a square free positive integer such that congruent to modulo . If we put into the , then but holds. Moreover, for the period of , we get

Let

be a modular automorphism of , then the fundamental unit of is given by the formulae

where is determined by , and .

Proof. Proof is omitted in ¹⁵.

Lemma 2.3. Let be a square free positive integer congruent to 2,3 modulo 4. If we put , into the , then we get , but .

Furthermore, for the period of , we have and

Besides, let

be a modular automorphism of . Then the fundamental unit of is given by the following formula:

where is determined by , and = for .

Proof. Proof can be obtained in a similar way as the proof of Lemma 2.2.

3. Theorems and Results

Theorem 3.1. Let be square free positive integer and be a positive integer.

(1) We suppose that

where is a positive integer. In this case, we obtain that and

with . Moreover, we get

for

(2) In the case of , if we assume that

for odd positive integer, then and

Also, in this case

hold for

Remark 1. it is clear that is odd number if In the case of (2), is not integer if we substitue odd positive numbers into the parametrization of for So, we have to put a condition as is divided by 3 in the case of (2). Also, if we choose is even integer, the parametrization of coincides with the case of (1). That's why, we have to consider and positive odd integer in the case of (2).

Proof. (1) For any and positive integer, holds since is odd integer. From Lemma 2.2, we know that and .

By using these equations, we obtain

So, we get

By a straight forward induction argument, we have

Using Definition 2.1 and rearanging the above equality, we obtain

This implies that since If we consider Lemma 2.2, we get

and

Now, we have to determine , and using Lemma 2.2 again. It is clear that by induction for If we substitute the values of sequence into the coefficients of fundamental unit

holds for

(2) In the case of , we get . By subsituting this equivalence into the parametrization of , we have for positive odd integer. By using Lemma 2.2 and the parametrization of , we have Then, we have

By a straight forward induction argument, we get

Rearranging and using Definition 2.1 into the above equality, we obtain

This implies that since If we consider Lemma 2.2, we get

and

Using for we obtain the coefficients of fundamental unit and for

Corollary 3.2. Let d be a square free positive integer congruent to 1 modulo 4. If we assume that is satisfying the conditions in Theorem 3.1, then it always hold Yokoi’s invariant =0.

Proof. is defined by Yokoi. In the case of (1) if we substitue and into the , then we obtain,

So, we get =0 since for positive integer.

In a similar way, we obtain

since for positive odd integer in the case of (2).

Corollary 3.3. Let be the square free positive integer positive integer corresponding to holding (1) in the Theorem 3.1. We tabulate the Table 3.1, where fundamental unit is , integral basis element is and Yokoi’s invariant is for and .

Proof. This Corollary is obtained from Theorem 3.1 by taking or 2 in the case of (1). is defined as If we substitue and into the , then we get

for Also, we get for . Since is increasing sequence, we obtain

for . So, we have for . Besides, in the case of we get for as well as for by using similar way. The proof of Corollary 3.3 is completed.

Remark 2. In the Table 3.1, using the classical Dirichlet class number formula, we calculate class number for both the real quadratic field and . These fields are also obtained in the Table 2.1 of reference ⁸. Additionaly, using the classical Dirichlet class number formula and computer calculations, we can see the other class numbers for several real quadratic fields in the Table 3.1 as follows:

Corollary 3.4. Let be the square free positive integer positive integer corresponding to holding (2) in Theorem 3.1. We state the Table 3.2 where fundamental unit is , integral basis element is and Yokoi’s invariant is for and

Proof. By subsituting or 3 into the (2) of Theorem 3.1, we get this corallary and the table. If we substitue and into the then we have

for Since is increasing sequence, we obtain

for . Therefore, we obtain for . Also, we get since

for and

Remark 3. We obtain class number as for the real quadratic field and for the real quadratic field in the Table 3.2 using the classical Dirichlet class number formula and computer calculations. Besides, we can see that other class numbers are too bigger than class number two by using Proposition 4.1 of reference ⁸.

Theorem 3.5. Let be a square free positive integer and be a positive integer such that . We assume that the parametrization of is

where is a positive integer. Then, following conditions hold:

(1) If and is even positive integer then

(2) If and is odd positive integer then .

Also, in real quadratic fields, we obtain

with for .

Furthermore, we have the fundamental unit and coefficients of fundamental unit as follows:

Proof. If we choose we obtain that is not integer because of the parametrization of

So, we have to consider that , in order to get .

(1) If we suppose that, then . Also, eitheror hold. By substituting these values into parametrization of by considering is even positive integer, we obtain . Moreover, if and is even positive integer, then and hold. By substituting these values into parametrization of , then we get .

(2) If and is odd positive integer then we get and Substituting these values into parametrization of and rearranging, we have .

By Lemma 2.3 we get

So, we have

By Lemma 2.3 we obtain

Using Definition 2.1 and put equation into the above equality, we have

This implies that since Let us consider Lemma 2.3, then we obtain

and hold.

Now, we can determine , and using Lemma 2.3 as follows:

and using the by induction for

Remark 4. We should say that the present paper has got the most general results for such type real quadratic fields. Moreover, we can obtain infinitely many values of which correspond to new real quadratic fields by using the results.

Corollary 3.6. Let be square free positive integer and be a positive integer satisfying that Suppose that the parametrization of is

Then ,we get and

with . Additionally, we get the fundamental unit , coefficients of fundamental unit and Yokoi’s invariant as follows:

Proof. If we put into the Theorem 3.5, then we get

and .

We have to calculate defined in the H.Yokoi’s references. If we substitue and into the , then we get

We obtain for For , we get

since is increasing sequence as well as the assumption. Therefore, we obtain for For numerical example, let us consider the following Table 3.3 where fundamental unit is , integral basis elemant is and Yokoi’s invariant is for

Remark 5. The class number is obtained for the real quadratic field and this field was got with same class number by Mollin in the reference ⁷ too. The real quadratic field has got class number in the Table 3.3 using the classical Dirichlet class number formula and computer calculations. Furthermore, we can not calculate easily class numbers for the other real quadratic fields since they are too bigger than class number two by using Proposition 4.1 of reference ⁸.

Corollary 3.7. Let be a square free positive integer and be a positive integer satisfying that . Suppose that the parametrization of is

Then, we have and

and Furthermore, we obtain following equalities for , and Yokoi’s invariant .

and ,

Proof. This Corollary is got by substituting into the Theorem 3.5.

We assume that and ,so we have

and

If we substitute and into the and rearranged, then we get

So, we have

since is increasing sequence. Therefore, we obtain for . To illustrate, let us consider the following Table 3.4 where fundamental unit is , integral basis elemant is and Yokoi’s invariant is for .

Corollary 3.8 Let be square free positive integer and be a positive integer satisfying that . We assume that the parametrization of is

Then, we get and

with . Besides, we obtain the fundamental unit , coefficients of fundamental unit and Yokoi’s invariant as follows:

and ,

Proof. We have this corollary by using Theorem 3.5 for It is just enough to calculate defined as If we substitue and into the , then we get

If we consider that is increasing sequence, we calculate following inequality

for

Hence, we obtain for

For numerical example, let us consider the following table where the fundamental unit is , integral basis elemant is and and Yokoi’s invariant is for

Remark 6. The class number is for the real quadratic field in the Table 3.5 using the classical Dirichlet class number formula and computer calculations. Furthermore, we can not calculate easily other class numbers since they are too bigger than class number two by using Proposition 4.1 of reference ⁸.

4. Conclusion

In this paper, we introduced the notion of real quadratic field structures such as continued fraction expansions, fundamental unit and Yokoi invariants where is square free positive integer. We established general interesting and significant results for that. Results obtained in this paper provide us a useful and practical method so as to rapidly determine continued fraction expansion of fundamental unit and and Yokoi invariants for such real quadratic number fields. There are some authors work on structure of the real quadratic number fields, but the results in this paper are new and more general for such types of real quadratic fields.

Findings in this paper will help the researchers to enhance and promote their studies on quadratic fields to carry out a general framework for their applications in life.

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