A computational technique for determining the fundamental unit in explicit types of real quadratic number fields

quadratic number field 𝑄(√𝑑) , integral basis element

> 1. The Unit Theorem for real quadratic fields says that every unit in the integer ring of a quadratic field is generated by the fundamental unit. Also, regulator in real quadratic cryptography is outstanding. We have seen that the regulator = log plays the role of a group order. The regulator problem is to find an integer ′ satisfies | ′ − | < 1 where ′ is an approximation of with any given precision can be computed in polynomial time for discriminant. However, some of the fundamental units can not be calculated by computer programme in short time because of the big numbers or long calculations of usual algorithm. This is also the main problem from the computing/informatics point of view. So, determining of the fundamental units is of great importance. In this paper, we construct a theorem to determine the some certain real quadratic fields (√ ) having specific form of continued fraction expansion of where ≡ 1( 4) is a square-free integer. We also present the general context and obtain new certain parametric representation of fundamental unit for such types of fields.By specialization, we get a fix on Yokoi's invariants and support all results with tables.

Keywords:
Quadratic fields Continued fractions Fundamental units Yokoi's invariants

Introduction
*Development of the theory of quadratic fields is not easy task. Tools of the infrastructure of quadratic fields have rendered numerous results in algebraic and computational number theory with cryptography as well as algebraic geometry, especially when applied to quadratic order. For example, Public Key Cryptography is one of the main techniques for making the internet secure in the cryptography and computer science. Most public key crypto-systems are based on intractable computational problems in number theory such as factoring integers. One source for computationally hard problems is algebraic number theory. Since the Diffie-Hellman key exchange protocol was presented in class groups of imaginary quadratic orders in 1988, many public key crypto-systems have been suggested whose security is based on difficult problems in quadratic number fields. Since then it has been started to state of the art of real and imaginary quadratic field crypto-systems.
Also, there are many different approach from that of most authors who use genus theory, composition of binary quadratic forms, and who use class field theory as a developmental tool. Also, many books and papers on the number theory include(use) continued fraction, ideal, class number, quadratic residue, prime producing quadratic polynomials, binary quadratic forms, elliptic curves, algorithms in cryptography based upon ideals with continued fraction algorithms, regulators in the class group, etc.
Because of the importance of the class number, the problem of determining the class number is of central interest in algebraic number theory. In general, the determination of the class number of an arbitrary algebraic number field is not an easy task. There exist some formulaes for determining the class numbers of real and imaginary quadratic fields, but the problem of determining the class number of an arbitrary number field is still beyond the scope of contemporary number theory.For example, to determine the class number (Dirichlet class number formulae), we need regulator, values of -function, discriminant etc. So, real quadratic fields have great importance in many branches of mathematics, even 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. Most of present and past works focused on the lower bound of fundamental units and the number of some special types of polynomials with fixed period of continued fraction expansions, certain class number and some types of continued fraction expansions, relations between the coefficients of fundamental units, comparison between the period length, quadratic fields and cryptology and Yokoi's invariants (Buchmann, 2004;Badziahin and Shallit, 2016;Benamar et al., 2015;Clemens et al., 1995;Elezovi´, 1997;Kawamoto and Tomita, 2008;Louboutin, 1988;Özer, 2016a;2016b;Sasaki, 1986;Tomita, 1995;Tomita and Yamamuro, 2002;Williams and Buck, 1994;Yokoi, 1990;1993a;1991;1993b;Zhang and Yue, 2014). For the history and main results on infrastructures of quadratic fields, we refer to the reader to (Mollin,1996;Olds, 1963;Perron, 1950;Sierpinski, 1964).
The focal point of this paper is to determine the some specific types of the real quadratic fields Q(√d) and the representation of fundamental units is a square free positive integer. By using this practical way, obtained results on fundamental units, Yokoi's invariants, continued fraction expansions, period length are given with tables as illustrates. Also, present paper completes (Özer, 2016b) in the case of d ≡ 1(mod4).

Prelimınaries
Now, we recall some definitions and lemmas which will be used later.

Quadratic fields
Definition 1. If is an extension of of degree two, then is called a quadratic field and represents as = Q(√d) where d is a square free integer.
Definition 2. If d > 0 square free integer, then Q(√d) is called a real quadratic field, and if d < 0 then Q(√d) is called a imaginary (complex) quadratic field.
Note 1. There is a one to one correspondence between quadratic fields and square free rational integer for d ≠ 1. Also, is called integral ring is the ring of integers of the quadratic field . The ring of integer of quadratic field has two integral basis elements. One of is the trivial identity element 1, another is the non trivial basis element . In real quadratic fields, is defined = 1+√d 2 in the case of d ≡ 1(mod4) and also = √d in the case of d ≡ 2,3(mod4).

Continued fraction expansions
There are many types of continued fraction expansions, but in our work, we use the quadratic irrational numbers and reduced quadratic irrationals, which indicate the periodic continued fraction expansion and purely periodic continued fraction expansion, respectively.

Fundamental units
Definition 7. Let (√ ) be a real quadratic number field and be a unit group. In real quadratic fields, positive units in have a generator, which is the smallest unit exceeds 1. This selection is unique and is called the fundamental unit of (√ ) and denoted by = 1 2 ( + √ ).
Note 2. When < 0, then unit group is finite cyclic and when > 0 then the positive units of form a multiplicative group isomorphic to , and so contains exactly one generator larger than 1 describing as fundamental unit. Proposition 1. Let (√ ) be a real quadratic number field, then there is a fundamental unit > 1 where the unit group of (√ ) is = {∓ | }. To illustrate to notion of fundamental unit, we have followings: Example 2. Let = 5, then the fundamental unit is Powers of (1 + √5) 2 ⁄ are also units and there are infinitely many of them since (1 + √5) 2 ⁄ > 1.
Remark 1. Not all fundamental units are so easy to calculate practically, even for small values of . So, this is very important to find a practical method so as to easily and rapidly determine fundamental unit . In this section we also give some fundamental concepts as follows for the proof of our main theorem defined in the next section.
Note 4. For the set ( ) of all quadratic irrational numbers in (√ ), we say that in ( ) is reduced if > 1,−1 < ′ < 0 ( ′ is the conjugate of with respect to ), and ( ) denotes 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 (√ ).

Main theorem and results
Main Theorem. Let be square free positive integer and ℓ ≥ 2 be a positive integer.
We can obtain following conclusions from Main Theorem.
Corollary 1. Let d be a square free positive integer congruent to 1 modulo 4. If we assume that is satisfying the conditions in Main Theorem, then it always hold Yokoi's invariant =0.

Conclusıon and future works
There are a lot of applications for quadratic fields in many different fields of mathematics which include algebraic number theory, algebraic geometry, algebra, cryptology, and also other scientific fields like computer science.
In this paper, we introduced the notion of real quadratic field structures such as continued fraction expansions, fundamental unit and Yokoi invariants. We established general interesting and significant results for that. Results obtained in this paper provide us a useful and practical method in order to rapidly determine continued fraction expansion of fundamental unit and and Yokoi invariants for such real quadratic number 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.
Future researches will be related to the application of our developed model/theory in crypto inteligent/smart systems.