SOME RESULTS ON SPECIAL CONTINUED FRACTION EXPANSIONS IN REAL QUADRATIC NUMBER FIELDS

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1 Journal of Mathematical Analysis ISSN: 7-34, URL: Volume 7 Issue 4(6), Pages SOME RESULTS ON SPECIAL CONTINUED FRACTION EXPANSIONS IN REAL QUADRATIC NUMBER FIELDS ÖZEN ÖZER, CENAP ÖZEL Abstract. The aim of this paper is to determine and investigate the continued fractions expansions of w d for the real quadratic number fields Q( d) for which the period has constant elements that are completely equal to (except the last digit of period) in the symetric part of the period of integral basis element where d, 3 mod 4 is a square free positive integer. Moreover, we give new explicit formulas for the fundamental unit ɛ d and Yokoi s d-invariants n d and m d in relation to continued fraction expansion of such form of w d. These new formulas are not known in the literature of real quadratic fields. Such types of real quadratic fields are classified as new results.. Introduction Let k = Q( d) be a real quadratic number field where d > is a positive squarefree integer. An integral basis element is denoted by w d = d for d, 3 mod 4 and l(d) is the period length in simple continued fraction expansion of algebraic integer w d = [ a ; a, a,..., a l(d), a ]. The fundamental unit ɛ d of real quadratic number field is also denoted by ) ɛ d = (t d + u d d / where N (ɛ d ) =( ) l(d). [[ ]] [[ ]] Moreover, Yokoi s invariants are expressed by n d = td u and m u d d = t d d where [[x]] represents the greatest integer not greater than x and also {G n } is a sequence which is defined in the sequel. There are a lot of applications for quadratic fields in many different fields of mathematics such as algebraic number theory, algebraic geometry, algebra, cryptology, and also other scientific fields like computer science. It is very important to determine fundamental unit and Yokoi s invariants in order to study on the class number problems and work on determination of the structures of real quadratic number fields. In recent years in [3] Tomita and his co-researcher shown a relation Mathematics Subject Classification. R, A55, R7. Key words and phrases. Quadratic Field, Continued Fraction, Fundamental Unit. c 6 Ilirias Publications, Prishtinë, Kosovë. Submitted April 8, 6. Published August 5, 6. 98

2 SPECIAL CONTINUED FRACTION EXPANSIONS 99 between real quadratic fields of class number one and a mysterious behaviour of the simple continued fraction expansion of certain quadratic irrationals. The theorems of Friesen in [] and Halter-Koch in [] have examined a construction of infinite families of real quadratic fields with large fundamental units. In [4], Tomita gave some results for fundamental unit by using Fibonacci sequence and continued fraction. Moreover, he determined continued fraction expansion of integral basis element for period is 3 in []. R.Sasaki and R.A.Mollin studied on lower bound of fundamental unit for real quadratic number fields and get certain important results in [] and in [5]. H.Yokoi defined two invariants important for class number problem and solutions of Pell equation by using coefficients of fundamental unit in [6]-[9]. Besides, first author in the paper obtained some special results for different forms of continued fraction expansion of w d in [7], [8] and [9]. For the getting more information about continued fraction expansions, we refer [], [] and [5]. In this paper, we are interested in the problem determining the continued fraction expansions which have [[ constant elements as s (except the last digit of the period, d ]] which is always for w d = d) for a given period length. There are infinitely many values of d having all s in the symmetric part of the period and here we will classify them with respect to arbitrary period length. We will also determine the general forms of fundamental units ɛ d as well as t d and u d which are the coefficients of fundamental units ɛ d = (t d+u d d) > by using this new formulization. The fundamental unit and Yokoi s invariant are easily calculated for d integers which seems to be the number of many figures. Finally, some results such as fundamental units, period length, Yokoi s invariants n d and m d which are defined in terms of coefficients of fundamental units are obtained with tables for main theorems which are mentioned in Section 3.. Preliminaries In this section we will begin by describing some fundamental concepts necessary to prove our main theorems in the next section. First we need to define a new sequence as follows. Definition. {G i } is called as a sequence defined by recurence relation G i = G i + G i for i, where G = and G =. Some values of this sequence can be found as follows: G = G +G =, G 3 = G +G = 4+ = 5, G 4 = G 3 +G = + =, G 5 = G 4 +G 3 = 4+5 = 9, G 6 = G 5 +G 4 = 7, G 7 = 69, G 8 = 48, G 9 = 985 G = 378, G = 574, G = 386, G 3 = 3346, G 4 = 878,... We will create our main results using this sequence. From [3] we have the following lemma.

3 ÖZEN ÖZER, CENAP ÖZEL Lemma.. For a square-free positive integer d congruent to, 3 modulo 4, let ω d = d, a = [ω d ], ω R = a + ω d. Then ω d / R(d), but ω R R(d) holds. Moreover for the period l = l(d) of ω R, we get ω R = [a, a,..., a l ] and ω d = [a, a,..., a l, a ]. Furthermore, let ω R = (P lω R +P l ) (Q l ω R +Q l ) = [a, a,..., a l, ω R ] ( d ) be a modular automorphism of ω R, Then the fundamental unit ɛ d of Q is given by the following formula and ɛ d = t d + u d d t d = a.q l(d) + Q l(d), = (a + d)q l(d) + Q l(d) u d = Q l(d) where Q i is determined by Q =, Q = and Q i+ = a i Q i + Q i, (i ). Lemma.. [[ Let d be a square free positive integer congruent to, 3 modulo 4 d ]] and a = denote the integer part of w d for d congruent to, 3 mod 4. If we suppose that w d has partial constant elements repeated s in the case of period l = l(d) then we have continued fraction expansions w d = d= [ ] [ ] a ; a, a,..., a l(d), a l(d) = a ;,,...,, a, wr = a t + d = [ a,,..., ] for quadratic irrational numbers and reduced quadratic irrational numbers respectively. Moreover, A j = a G j+ + G j and B j = G j+ are determined in the continued fraction expansions where {A j } and {B j } are two sequences defined by A =, A =, A j = a j A j + A j, B =, B =, B j = a j B j + B j, for j and j < l(d) where l(d) is the period length of w d. Then C j = A j /B j is the j th convergent in the continued fraction expansion of d, so that A l = a G l + 3a G l +G l and B l = a G l +G l for j = l(d). Furthermore, in the continued fraction w R = a + d = [b, b,..., b n,... ] = [a,,...,,...], P k = a G k + G k and Q k = G k are determined in the continued fraction expansion where {P k } and {Q k } are two sequences defined by P =, P =, P k+ = b k+.p k + P k, for k. Q =, Q =, Q k+ = b k+.q k + Q k, Proof. We prove this by induction on k. Using the following table including the values of A k, B k and a k we can determine the convergence of w d = [ a ;,,...,, a ] for l(d) > 4. So, we can easily say that this is true for k =.

4 SPECIAL CONTINUED FRACTION EXPANSIONS k a k a... A k a G a G + G a G 3 + G a G 4 + G 3 a G 5 + G 4... (a ) (a + ) (5a + ) (a + 5) (9a + )... B k Table G G G 3 G 4 G 5... Now, we suppose that the result is true for k < i and < i l. Using the relations of the sequence {G i }, we obtain A k+ = a k+.a k + A k = (a G k+ + G k ) + (a G k + G k ) = a (G k+ + G k ) + (G k + G k ) = a G k+ + G k+. We can also get the following result as follows B k+ = a k+.b k + B k = G k+ + G k = G k+. Moreover, since a l = a, we obtain A l = a G l + 3a G l + G l and B l = a.g l + G l for k = l(d) in an easy way. Furthermore, in the continued fraction a + d = [b, b,..., b n,... ] = [a,,...,,...], we obtain the following table k b k a... P k a G + G a G + G a G 3 + G a G 4 + G 3 a G 5 + G 4... (a ) (4a + ) (a + ) (4a + 5) (58a + )... Q k G G G G 3 G 4 G 5... Table.. This completes the proof. Definition. Let c n = a.c n + b.c n be a recurrence relation of the sequence {c n } where a, b are real numbers. Then the solution depends on the nature of the roots of the characteristic equation r ar b =. Hence, we can find the characteristic equation as r r = for the sequence {G k }. Therefore, we can write each element of the sequence as follows. G k = [ ( + k ( ) ) ] k for k.

5 ÖZEN ÖZER, CENAP ÖZEL Remark. Let {G n } be the sequence defined as in Definition. Then, we state the following: mod 4, if n, 3 mod 4; G n mod 4, if n mod 4; mod 4, if n mod 4 for n. 3. Main results The following are our main theorems and results with the notations of the previous section. Theorem 3.. Let d be a square free positive integer and l be a positive integer. Suppose that parametrization of d is d = s G l + s(g l + G l ) +. for all integers s >. If s is a even positive integer then d mod 4 holds and we have w d = sg l + ;,,...,, + sg }{{} l l with l = l(d). Also, the fundamental unit ɛ d, coefficients of fundamental unit t d, u d are in the following ɛ d = (sg l + ) G l + G l + G l d, t d = (sg l + ) G l + G l and u d = G l. Proof. If l is an arbitrary positive integer greater than and s is an even positive integer then we obtain d mod 4 since d = s G l + s(g l + G l ) +. If we substitute w d into the w R, we get w R = (sg l + ) + sg l + ;,,...,, sg }{{} l + l and w R = (sg l + ) w R = (sg l + )+ + + w R. By Lemma. and Lemma. with the properties of continued fraction expansion, we get w R = (sg l + ) + G l w R + G l G l w R + G l. Moreover, using Definition into the last equality, we obtain w R (sg l + ) w R ( + sg l ) =.

6 SPECIAL CONTINUED FRACTION EXPANSIONS 3 This requires that w R = (sg l + ) + d since w R >. If we consider Lemma., we obtain and l = l (d). w d = d = sg l + ;,,...,, sg }{{} l + l Now, we should determine ɛ d, t d and u d using Lemma., we can get easily Q = = G, Q = a.q + Q Q = = G Q 3 = a Q + Q = G + G = 5 = G 3, Q 4 = = G 4. So, this implies that Q i = G i by using mathematical induction for il. If we substitute these values of the sequence into the ɛ d = t d+u d d = (a + d)q l(d) + Q l(d) and rearrange, then we get the desired result ɛ d = (sg l + ) G l + G l + G l d, t d = (sg l + ) G l + G l and u d = G l. Corollary 3.. Let d be a square free positive integer and l be a positive integer satisfying that l. Suppose that parametrization of d is Then, we have d mod 4 and d = 4G l + (G l+ + G l ) +. w d = G l + ;,,...,, + 4G }{{} l l with l = l(d). Additionally, we can get fundamental unit ɛ d, coefficients of fundamental unit t d, u d as follows: ɛ d = (G l + ) G l + G l + G l d, t d = (G l + ) G l + G l and u d = G l. Also, we have Yokoi s d- invariant value n d = for all l. Proof. This claim is obtained if we substitute s = into the Theorem 3.. Now we have to prove that Yokoi s d- invariant value is n d = for all l. [[ ]] We know from H. Yokoi s references [6],[7], [8], [9] that n d = td. If we u d substitute t d and u d into n d, then we get [[ ]] [[ td 4G ]] [[ n d = u = l + G l + G l d 4G = + + G ]] l l G l G =, l since (G l ) is increasing and < G l + G l, 375 for l. Therefore, we G l obtain n d = for l owing to definition of n d. This completes the proof of the corollary.

7 4 ÖZEN ÖZER, CENAP ÖZEL For numerical example, let us consider the following table where fundamental unit is ɛ d, integral basis element is w d and Yokoi s invariant is n d for l(d) (In this table, we will rule out l(d) = 6 since d is not a square free positive integer in this period). d l(d) n d w d ɛ d 3 [5;, ] [;,, ] [5;,,, 5] [59;,,,, 8] [339;,,,,,, 678] [87;,,,,,,, 634] [97;,,,,,,,, 394] [4757;,,,,,,,,, 954] [483;,,,,,,,,,, 966] Table 3.. Theorem 3.3. Let d be a square free positive integer and l be a positive even integer. We assume that parametrization of d is d = s G l + (G l + G l ) s +. 4 for any positive integer s. Then we have the following. () If l mod 4 and s mod 4 positive integer then d 3 mod 4 holds. () If l mod 4 and s mod 4 positive integer then d mod 4 holds. Also we have w d = sg l + ;,,...,, sg }{{} l + l and l = l(d). Moreover, we have following equalities ( ) sg ɛ d = l + G l + G l + G l d for ɛ d, t d and u d. t d = sg l + G l + G l and u d = G l Proof. We assume that l mod. Then we have two distinct cases. Now we can do the analysis of two cases as follows. () If l mod 4 holds then we have G l mod 4, G l mod 4. If we consider positive integer s mod 4 and substitute these equivalency and equations into the parametrization of d, then we have d 3(mod4). () If l mod 4 holds, then we have G l mod 4, G l mod 4. Therefore we get d mod 4 using positive integer s mod 4 and the above equivalence. So, we obtain d mod 4.

8 SPECIAL CONTINUED FRACTION EXPANSIONS 5 By using Lemma., we have so we will gett w R = (sg l + ) + w R = sg l + + sg l + ;,,...,, sg }{{} l +, l + + = (sg l + ) w R w R By Lemma. and Lemma. with the properties of continued fraction expansion, we obtain w R = (sg l + ) + G l w R + G l G l w R + G l, by induction. By applying Definition into the above equality, we obtain w R (sg l + ) w R ( + sg l ) =. This requires that w R = sg l + + d since w R >. If we consider Lemma., we get w d = d = w d = sg l + ;,,...,, sg }{{} l + l and l = l (d). This shows that part of the first proof is completed. Now, we should determine ɛ d, t d and u d using Lemma., we get Q = = G, Q = a.q + Q Q = = G, Q 3 = a Q + Q = G + G = 5 = G 3, Q 4 = = G 4,... So, this implies that Q i = G i by using mathematical induction for i. If we substitute these values of the sequence into the ɛ d = t d+u d d = (a + d)q l(d) + Q l(d) and rearrange, we have ( ) sg ɛ d = l + G l + G l + G l d t d = sg l + G l + G l and u d = G l for ɛ d, t d and u d. This completes the proof of Theorem 3.3. Corollary 3.4. Let d be the square free positive integer and l be a positive integer holding that l mod and l >. We assume that parametrization of d is ( ) d = G l 4 + Gl+ + G l +.

9 6 ÖZEN ÖZER, CENAP ÖZEL Then, we get d, 3 mod 4 and w d = G l + ;,,...,, G }{{} l + l and l = l(d). Moreover, we have following equalities: ( ) G ɛ d = l + G l + G l + G l d for ɛ d, t d, u d and Yokoi s invariant m d. t d = G l + G l + G l and u d = G l {, if l = ; m d = 3, if l 4 Proof. This corollary is obtained if we substitute s = into the Theorem 3.3. We should show that {, if l = ; m d = 3, if l 4.. If we put t d and u d into the m d and rearranged, then we obtain [[ ]] [[ u m d = d 4G ]] l = t d G l + G l + G l By using the above equality, we have m d = for l =. From the assumption since (G l ) is increasing we get, ( 4 > G ) l > 3, 35 G l [[ 4G for l 4. Therefore, we obtain m d = l definition of m d and we can describe m d as follows: {, if l = ; m d = 3, if l 4. This completes the proof of the corollary. G l G l +G l+g l ]] = 3 for l 4 due to To illustrate the corollary, let us consider the following table where fundamental unit is ɛ d, integral basis element is w d and Yokoi s invariant is m d for < l(d) 8 (In this table, we will rule out l(d) =,, 4, 6 since d is not a square free positive integer in these periods). d l(d) m d w d ɛ d 6 [;, 4] [7;,, 4] [36;,,,,, 7] [5;,,,,,,, 4] [376;,,,,,,,,,,,,,,,,, 744] Table 3..

10 SPECIAL CONTINUED FRACTION EXPANSIONS 7 Remark. We should say that the present paper has got the most general theorems for such type real quadratic fields. Also, we can obtain infinitely many values of d which correspond to Q( d) and determine structures of such fields by using our theorems. References [] C.Friesen, On continued fraction of given period, Proc. Amer. Math. Soc.,, :9-4, 988. [] F.Halter-Koch, Continued fractions of given symmetric period, Fibonacci Quart., 9(4), 98-33, 99. [3] F.Kawamoto and K.Tomita, Continued fraction and certain real quadratic fields of minimal type, J.Math.Soc. Japan, 6, , 8. [4] S.Louboutin, Continued Fraction and Real Quadratic Fields, J.Number Theory, 3, 67-76, 988. [5] R. A. Mollin. Quadratics, CRC Press, Boca Rato, FL., 996. [6] C. D. Olds. Continued Functions, New York: Random House, 963. [7] Ö. Özer. On Real Quadratic Number Fields Related With Specific Type of Continued Fractions, Journal ofanalysis and Number Theory, Vol.4, No., 85-9, 6. [8] Ö. Özer. Notes On Especial Continued Fraction Expansions and Real Quadratic Number Fields, Kirklareli University Journal of Engineering and Science, Vol., No:, 74-89, 6. [9] Ö. Özer. Fibonacci Sequence and Continued Fraction Expansion in Real Quadratic Number Fields, Malaysian Journal of Mathematical Science, 6.(In press) [] O. Perron. Die Lehre von den Kettenbrchen, New York: Chelsea, Reprint from Teubner,Leipzig,99., 95. [] R. Sasaki. A characterization of certain real quadratic fields, Proc. Japan Acad., 6, Ser. A, no. 3, 97-, 986 [] W. Sierpinski. Elementary Theory of Numbers, Warsaw: Monografi Matematyczne, 964. [3] K. Tomita. Explicit representation of fundamental units of some quadratic fields, Proc. Japan Acad., 7, Ser. A, no., 4-43, 995. [4] K. Tomita and K. Yamamuro. Lower bounds for fundamental units of real quadratic fields, Nagoya Math. J., Vol.66, 9-37,. [5] K. S. Williams and N. Buck. Comparison of the lengths of the continued fractions of ( D and + ) D, Proc. Amer. Math. Soc., no. 4, 995-, 994. [6] H. Yokoi. The fundamental unit and class number one problem of real quadratic fields with prime discriminant, Nagoya Math. J.,, 5-59, 99. [7] H. Yokoi. A note on class number one problem for real quadratic fields, Proc. Japan Acad., 69, Ser. A, -6, 993. [8] H. Yokoi. The fundamental unit and bounds for class numbers of real quadratic fields, Nagoya Math. J., 4, 8-97, 99. [9] H. Yokoi. New invariants and class number problem in real quadratic fields, Nagoya Math. J., 3, 75-97, 993. Özen ÖZER Department of Mathematics Faculty of Science and Arts, Kırklareli University, 39-Kırklareli, Turkey. address: ozenozer39@gmail.com Cenap ÖZEL Department of Mathematics Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Department of Mathematics Faculty of Science, Dokuz Eylul University, 356 Izmir Turkey address: cenap.ozel@gmail.com