A Study on the Fundamental Unit of Certain Real Quadratic Number Fields

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.


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 ≡ 2,3 ( 4). 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 = ( + √ )/2 > 1 of the ring of algebraic integers in a real quadratic number field ℚ�√ � 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 is the period length of continued fraction expansion.
For the set ( ) of all quadratic irrational numbers in ℚ�√ �, we say that in ( ) is reduced if > 1,−1 < ′ < 0 ( ′ is the conjugate of with respect to ℚ ), 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 ℚ�√ �. Also, Yokoi's invariants, which were defined by H.Yokoi are determined by the coefficient of fundamental unit 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 ℚ�√ � real quadratic number fields.
The aim of this paper is to classify some types of ℚ�√ � 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 ≡ 2,3( 4) but also in the case of ≡ 1( 4)). 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.

Prelimineries
In this section we also give some fundamental concepts for the proof of our main theorems defined in the next section.
Proof is omitted in [15]. Lemma 2.3. Let be a square free positive integer congruent to 2,3 modulo 4. If we put = √ , 0 = �√ � into the = 0 + , then we get ∉ ( ), but ∈ ( ). Furthermore, for the period = ( ) of , we have = � 2 0 , 1 , 2 , … , ( be a modular automorphism of . Then the fundamental unit of ℚ�√ � is given by the following formula: Proof can be obtained in a similar way as the proof of Lemma 2.2.

Theorems and Results
Theorem 3.1. Let be square free positive integer and ℓ ≥ 2 be a positive integer.
is defined as If we substitue and into the , then we get Also, we get = 3 for ℓ = 2 . Since ℓ is increasing sequence, we obtain 2 1 2 2 9 2 2,113 2 for ℓ ≥ 3. So, we have = 2 for ℓ ≥ 3. Besides, in the case of = 2, we get = 5 for ℓ = 2 as well as = 4 for ℓ ≥ 3 by using similar way. The proof of Corollary 3.3 is completed.
(1) If we suppose that ℓ ≡ 1,2 ( 6) , then ℓ ≡ 1( 4) . Also, either ℓ−1 ≡ 0( 4) or ℓ−1 ≡ 1( 4) hold. By substituting these values into parametrization of by considering is even positive integer, we obtain ≡ 2,3 ( 4) . Moreover, if ℓ ≡ 4( 6) and is even positive integer, then ℓ ≡ 3( 4) and ℓ−1 ≡ 2( 4) hold. By substituting these values into parametrization of , then we get ≡ 3( 4). (2) If ℓ ≡ 5 ( 6) and is odd positive integer then we get ℓ ≡ 1( 4) and ℓ−1 ≡ 3( 4). Substituting these values into parametrization of and rearranging, we have ≡ 2( 4 = (2 + 1) ℓ 2 + 9 ℓ + 2 ℓ−1 and = 2 ℓ using the = by induction for ∀ ≥ 0. 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 ℚ�√ � by using the results. Corollary 3.6. Let be square free positive integer and ℓ ≥ 2 be a positive integer satisfying that ℓ ≢ 5( 6), 3 ∤ ℓ Suppose that the parametrization of is Then ,we get ≡ 2,3( 4) and 1 9 ;9,9, ,9,9 2 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 2 ≤ ℓ( ) ≤ 13. Remark 5. The class number is obtained ℎ = 1 for the real quadratic field ℚ�√83� and this field was got with same class number by Mollin in the reference [7] too. The real quadratic field ℚ�√142967� has got class number ℎ = 12 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 ℓ > 1 be a positive integer satisfying that ℓ ≡ 5 ( 6). Suppose that the parametrization of is 0 0, 253 4 Hence, we obtain 2 1 2  Remark 6. The class number is ℎ = 128 for the real quadratic field ℚ�√3504795� 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].

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.