Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields

Transcription

1 Malaysian Journal of Mathematical Sciences (): 97-8 (07) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields Özen Özer Department of Mathematics, Faculty of Science and Arts, Kırklareli University, Kırklareli, Turkey Received: 8th April 06 Accepted: 5th November 06 ABSTRACT In 00, Tomita and Yamamuro defined several theorems for fundamental unit of certain real quadratic number fields. Although, there are infinitely many values of having all s in the symmetric part of continued fraction expansion of, Tomita and Yamamuro (99) had described explicitly one type of for the fundamental units of the real quadratic fields by using Fibonacci sequence in the Theorem 3 for,3 4 and in the Theorem in the case of 4 (00). The main purpose of this paper is to generalize and provide an improvement of the theorem 3 and the theorem in the paper of Tomita and Yamamuro (00). Moreover, the present paper deals with new certain formulas for fundamental unit and Yokoi s -invariants, in the relation to continued fraction expansion of for such real quadratic fields. All results are supported by numerical tables. 00 Mathematical Subject Classification: B39, R, A55, R7. Keywords: Fibonacci Sequence, Quadratic Field, Continued Fraction, Fundamental Unit, Yokoi s invariants.

2 Özen Özer. Introduction Let = Q be a real quadratic number field where > 0 is a positive square free integer. In real quadratic fields, integral basis element is denoted by = = ;############################,,, l!", $ for,3 4 and = % = ;#################################,,, l!", $ for 4 where ld is the period length in simple continued fraction expansion of algebraic integer w d The fundamental unit ε d of real quadratic number field is also denoted by = ( )%* ) > where +, ε d - = l!. Determining the fundamental units of real quadratic fields is of great importance in the class number problem and the unit theorem since fundamental unit generates the unit group of real quadratic fields. 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. The theorem of Friesen (988) and Halter-Koch (99) was examined a construction of infinite families of real quadratic fields with large fundamental units. Kawamoto and Tomita (008) were determined minimal type of continued fraction for certain real quadratic fields. Also, Zhang and Yue (04) proved some congruences about the coffecient of fundamental unit. Benamar et al. (05), considered the real numbers which their quotient s elements are / is the largest power of dividing 0 +. By using coefficients of fundamental unit, Yokoi (990, 99, 993) defined two significant invariants as = * ) 3 4 and ( = 5 ( ) ) * 36 where 789 represents the ) greatest integer not greater than 8 for class number problem and the solutions of Pell equation. Lower bound of fundamental unit studied by Sasaki (986) and Mollin (996). = ( )% * ) > of = Q was Originally, we examine the continued fraction expansions which have partial quotient elements repeated as s in the period length for where is a square free integer. Although there are infinitely many values of having all s in the symmetric part of the period of continued fraction expansion of integral basis element, Tomita and Yamamuro (00) described explicitly one type of s in their paper for determining the fundamental units of the real quadratic fields. 98 Malaysian Journal of Mathematical Sciences

3 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields We get infinitely many new real quadratic fields which have got the form of = ; ;,,, <=>=? or = ; ;,,, ###################### where,3 4 or 4 respectively for the parametrization of square free positive integer. In this paper, we also classify the real quadratic fields in the main theorems according to arbitrary period length. For the classifications,we also determined the general forms of fundamental units and coefficents of fundamental units A, B in the terms of fibonacci sequence. Then, we get a fix on Yokoi s invariants and support all results with tables by the specialization. Also, we should say that the results obtained in this paper useful in the literature of class field theory.. Preliminaries We need following definitions and lemmas which are useful for results in the next section. Definition.. CD / E is called as Fibonacci sequence if it is defined by the recurrence relation, D / = D /" +D /" for 0, with seed values D = 0 and D =. Note. For the set G of all quadratic irrational numbers in Q, we say that H in G is reduced if H >, < H < 0, (H J is the conjugate of H with respect to Q), and denote by K the set of all reduced quadratic irrational numbers in G.Then, it is well known that any number H in K is purely periodic in the continued fraction expansion and the denominator of its modular automorphism is equal to fundamental unit of Q. Lemma.. Let be a square-free positive integer such that congruent to % modulo 4. If we put =, = L M into the N = +, then K but N Kholds. Moreover, for the period Q = ld of N, we get N = L #########################Mand,,..., R" = L, #########################].,..., R", Let N = S TU V %S TWX Y T U V %Y TWX = L,,..., R", N ##############################M be a modular automorphism of N, then the fundamental unit Z of [ is given by the formulae Malaysian Journal of Mathematical Sciences 99

4 Özen Özer = A + B A =.[ l + [, B = [ l where [ / is determined by [ = 0,[ = and [ /% = / [ / + [ /" 0. Proof. Proof is in Tomita (995). Lemma.3. For a square free positive integer congruent to,3 modulo 4, we put = and = \ ] into the N = +. Then K, but N K holds. Moreover, for the period Q = Q of N, we get and N = ^ <======>======? ############################,,,, R" _ l = ;############################,,, R", $. Furthermore, N = U VS T %S TWX = U V Y T %Y TWX,,,, R", N $ be a modular automorphism of N. Then the fundamental unit of [ is given by the following formula: = A + B = + [ l +[ A = [ l +[ and B = [ l where [ / is determined by [ = 0, [ = and [ /% = / [ / +[ /" 0. Proof. It can be proved in a similar way of previous lemma. Lemma.4. Let,3 4 be a square free positive integer and = \ ] denote the integer part of = for,3 4.If we consider which includes partial constant elements repeated s in the case of period Q = Q, then we have the continued fraction expansion as the form of 00 Malaysian Journal of Mathematical Sciences

5 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields = = ;###############################,,, l!", l! $ = L ;,,,, ################ M N = + = L ###############,,,,M for integral basis element and reduced integral basis element respectively. Furthermore, `a = D a% +D a and b a = D a% are determined in the continued fraction expansion of where C`aE and Cb a E are two sequences defined by : `" = 0,`" =, `a = a`a" +`a" b " =, b " = 0, b a = a b +b a" for 0 l and `d = D l! +3 D l!" +D l!" b = D l! +D l!" for = ld where Q = ld is period length of = and e d = `d f bd is the g (h convergent in the continued fraction expansion of. Moreover, i d = D d +D d" and [ d = D d are determined in the continued fraction Lj,j,j k,j l, M = L,,,,, M, where mi d n and m[ d n are two sequences defined by: i " = 0,i =, i d% = j d% i d +i d" and for g 0. [ " =, [ = 0, [ d% = j d% [ d +[ d" Proof. We can prove by using mathematical induction. Using the following table which includes values of `a, b a and a, we can easily say that this is true for = 0. Malaysian Journal of Mathematical Sciences 0

6 Özen Özer Table : (Converge of L ;,,,, ################ M for Q = Q) o p o (5 +3 q o 0 D +D D +D D k +D D r +D k D s +D r t o 0 D D D D k 3 D r 5 D s Now, we assume that the result true for < 0 and 0 Q. Using the defined the fibonacci sequence, we obtain ( / = for 0 Q ) A = a A + A = D a% +D a + D a +D a" k+ k+ k k = D a% +D a + D a +D a" = D a% +D a% B = a B + B = D a% +D a = D a% k + k + k k Moreover, since R = we get the following result : A = a F + 3a F + F l 0 l 0 l l Bl = a0fl + Fl ( for = Q ) Furthermore, in the continued fraction Lj,j,j k,j l, M = L,,,,, M, we have following table and this completes the proof. 0 Malaysian Journal of Mathematical Sciences

7 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields Table : (Converge of L,,,,, M) o v o w o 0 D +D + D +D 4 + D k +D 6 + D r +D k y o 0 D D D D k 3 D r Note: By defining similar new lemma, we can also obtain similar tables in the case of = % for 4. Remark.5. We can also write characteristic equation for Fibonacci sequence as the form of for 0. D a = 5 { + 5 } a 5 a } ~ Remark.6. Let CD E be Fibonacci sequence. Then, we state the following: for n 0. D 0 4 ; ;,,5 6 4 ; ; Main Theorems and Results Main theorems and their results will be given with the notation of the preliminaries section as follows. Theorem 3.. (Main Theorem ). Let be square free positive integer and l be a positive integer satisfying that 3 l. Suppose that the parametrization of is Malaysian Journal of Mathematical Sciences 03

8 Özen Özer = H+D l + } +H +D + where H 0 is a positive integer. Then following conditions hold: () If l 6 and H is even positive integer then 4 holds. () If l 6 and H is even positive integer then 3 4 holds. (3) If l 4 6 and H is even positive integer then 3 4 holds. (4) If l 5 6 and H is odd positive integer then 4 holds. In the Q real quadratic fields, we have = ; ƒ% l% for,3 4. ;,,, <=>=? ###############################,H +D l and l = l Additionally, we get the general form of fundamental unit and coefficients of fundamental unit A,B as follows: = ƒ% l% + D l +D, A = H +D l +D l +D and B = D l Proof. It is clear that % for all l 0 3 since Remark.6. We assume that 3 l,l in order to get %. Originally, we have to show that four conditions hold as the followings: () If l 6, then D l 4 and D 0 4 hold. If we consider that H is any even positive integer, then we obtain 4 by substituting these equivalences into the parametrization of. () If we consider l 6, then we haved l 4 and D 4. By substituting these values into parametrization of and rearranging, we obtain 3 4 for any H even positive integer. (3) If l 4 6, then we have D l 3 4 and D 4. By substituting these equivalences into the parametrization of, we get 3 4 where H is any even positive integer. (4) If l 5 6 and H is odd positive integer then we get D l 4 and D 3 4. So, we obtain 4. Hence, conditions are satisfied. 04 Malaysian Journal of Mathematical Sciences

9 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields By using Lemma.3 and Lemma.4, we get N = H+D l + }+; H +D l + ;,,, ############################### <=>=?,H+D l N = H +D l = H +D l N + + N Using Lemma.3 and the properties of continued fraction expansions, we have N = H +D l ++ D N +D D l N +D If we rearrange the above equation, we obtain N H +D l + N +H +D = 0 This implies that N = ƒ% l% + since N > 0. If we consider Lemma.3 and Lemma.4, we get = ; ƒ% l% ;,,, <=>=? ###############################,H +D l and l = l. This shows that = ; ƒ% l% first part of proof is completed. ;,,, <=>=? ###############################,H +D l holds. So, the Now, we should determine, A and B using Lemma.3. We can get easily following equations: Q = 0 = D, Q = = D, Q =. Q + Q Q = = D Q k = Q + Q = D +D =D k, Q r = D r, Malaysian Journal of Mathematical Sciences 05

10 Özen Özer So, this implies that [ / = D / by using mathematical induction for 0 0. If we substitute these values of sequence into the = A + B = + [ l +[ and rearranged, we have = ƒ% l% + D l +D, A = H +D l +D l +D and B = D l which completes the proof of the Main Theorem. Remark 3.. Theorem 3. includes the parametrization of depends on the arbitrary H positive integer parametre. The case that H=0 was tread in Tomita and Yamamuro (00), but we should say that the present paper has got the most general theorem and results for such type of the real quadratic fields. Also, we can get infinitely many values of which correspond to new Q for H by using our results. We obtain following results on Yokoi s invariants as well as fundamental unit and continued fraction expansion. Corollary 3.3. Let be square free positive integer and l be a positive integer satisfying that l 5 6. Suppose that parametrization of is =, +3D l - +3D + then we have 4 and = ; %k l ;,,, ######################@ <=>=?,+3D l with l = l. Additionally, we get the fundamental unit, coefficients of fundamental unit A,B and Yokoi s invariant as follows: = %k l + D l +D, A = 3D l +D l +D and B = D l = Besides, we state the following Table 3 where fundamental unit is, integral basis elemant is and and Yokoi s invariant is for 5 l Malaysian Journal of Mathematical Sciences

11 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields Table 3: Square-free positive integers d with 5 l 3. Œ Œ Ž Œ Œ Œ 74 5 L8;,,,,6 #############M L34;,,,,68 ################M L396;,,,,479 ##################M ;,,,,8597 ###################$ Proof. We create this result by subsituting H = into the Main Theorem. We have to determine value of. We know that is defined as = * ) 3 ( ) 4 in Yokoi s references. If we substitue A and B into the, then we get 4D l = B = A 3D l +D l +D Using assumption and the above equality, we get 4D l > 3D } >,6 l +D l +D since Fibonacci sequence is monotone increasing. Therefore, we obtain = r l 3 5 k 3 6 =. This completes the proof of Corollary 3.3. l % l % lwx Also, Table 3 which can be extended for the different values of l is given numerical examples for Corollary 3.3. Corollary 3.4. Let be the square free positive integer and l > be a positive integer such that 3 l,l 5 6. We assume that the parametrization of is =, +5D l - +5D + then we get,3 4 and = ; %s l ;,,, <=>=? ######################@,+5D l and l = l. Malaysian Journal of Mathematical Sciences 07

12 Özen Özer Moreover, we have following equalities : =, +5D l -D l +D + D l A = 5D l +D l +D and B = D l for, A,B and Yokoi s invariant. = ;0š l = ;0š l > Besides, we state the following Table 4 where fundamental unit is, integral basis elemant is and Yokoi s invariant is for < l 3. (In the following table, we rule out l = 4,8,0 since d is not a square free positive integer in these periods). Table 4.: Square-free positive integers d with l 3. Œ Œ Œ Œ Œ 5 L3;,6 ####M 30 7 L33;,, ##############M, L538;,, #################M, Proof. Corollary is obtained if we subsitute H = into the Main Theorem. So, we have to determine the value of Yokoi d-invariant. If we substitute A and B into the and rearranged, then we obtain = A B = 5D l +D l +D 4D l If we put l = above equation, we get =. For l >, by using = 5 ( ) * ) 3 6 = + 5 r + r l + lwx l 36 and considering that the Fibonacci sequence is monotone increasying, we obtain 08 Malaysian Journal of Mathematical Sciences

13 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields 0 < D l +D l +D 4D l < 0,93 Hence, we determine the value of Yokoi s invariant as = 5 s l 3 % l % lwx r l 3 6 =. Besides, Table 3. is given as an illustrate of this corollary. Corollary 3.5. Let be square free positive integer and l be a positive integer satisfying that l 5 6. If we choose the parametrization of as the form of =, 7D l D + then 4 and = ; œ l% ;,,, <=>=? ####################@ 7D l + with l = l. Moreover, we get the fundamental unit, coefficients of fundamental unit A,B and Yokoi s invariant as follows: = œ l% + D l +D, A = 7D l +D l +D and B = D l = Furthermore, we state the following Table 5 where fundamental unit is, integral basis elemant is and and Yokoi s invariant is for 5 l 3. Table 5: Square-free positive integers d with 5 l 3. Œ Œ Œ Œ Œ L8;,,,36 ########## M L3;,,,,64 ################ M L5590;,,,80 ################# M L00300;, ##################,00600M Malaysian Journal of Mathematical Sciences 09

14 Özen Özer Proof. This claim is obtained if we subsitute H = 3 into the Main Theorem. It is sufficient to determine Yokoi s invariant value of = 5 ( ) * ) 3 6. If we substitue A and B into the, then we get = A B = 7D l +D l +D 4D l Since CD l E is monotone increasing, we get following inequality < 7D l +D l +D 4D l,86 for l. Therefore, we obtain = 5 œ l 3 % l % lwx r l 3 6 = which completes the proof of Corollary 3.5. For the numerical examples, we give the Table 5. Theorem 3.6. (Main Theorem ) Let be square free positive integer and l be a positive integer. () We suppose that = HD l + +8HD +4 where H > 0 is a positive integer. In this case, we obtain that d mod4 and with l = l. Moreover, we get = ;HD l +;,,, #######################@ <=>=?,+HD l for = ( )%* ). A = HD l +D l +D and B = D l () In the case that l 0 3, if we assume that = HD l + +4HD +4 for H > 0 positive odd integer, then d mod4 and = ;+ ƒ l ;,,, ######################@. <=>=?,+HD l 0 Malaysian Journal of Mathematical Sciences

15 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields Also, in this case A = HD l +D l +D and B = D l hold for = ( )%* ). Remark 3.7. it is clear that D l is odd number if l 0 3 holds. ƒ l is not integer if we substitue H odd integer into the papametrization of d fin the case of l 0 3. So, we assume that l is divided by 3 in the case of (). Also, if we choose H is even integer, the parametrization of coincides with the case of (). That's why we assume l 0 3 and H > 0 positive odd integer in the case of (). Proof. () For any l and H > 0 positive integer, 4 holds since HD l + is odd integer. From Lemma., we know that = and N = +. By using these equations, we obtain %, = L M N = HD l +;HD l +;,,, #######################@ <=>=?,+HD l N = +HD l = +HD l U V By a straight forward induction argument, we have + + N N = +HD l + + D N +D D l N +D Using Definition. into the above equality, we obtain N +HD l N +HD = 0 Malaysian Journal of Mathematical Sciences

16 This implies that., we get % Özen Özer N = ƒ l%% since N > 0. If we consider Lemma = ;+HD l ;,,, <=>=? #######################@,+HD l and l = l. Proof of the first part of () is completed. Now, we have to determine, A and B using Lemma.. It is easily seen that [ / = D / 0 0 by induction. If we substitute the values of sequence into the coefficients of fundamental unit holds for = A +B. A = HD l +D l +D and B = D l () In the case of l 0 3, we get D l 0. By subsituting this equivalence into the parametrization of, we have 4 for H > 0 positive odd integer. By using Lemma. and the parametrization of = HD l + +4HD + 4, we have N = + N = +HD l + N = ƒ l +;+ƒ l ;,,, ######################@ <=>=?,+HD l % XŸ X X = +HD l Ÿ X XŸ X V U V By a straight forward induction argument, we get N = +HD l + D N +D D l N +D Using Definition. into the above equality, we obtain N +HD l N +HD = 0 This implies that., we get N = ƒ l + % since N > 0. If we consider Lemma Malaysian Journal of Mathematical Sciences

17 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields % = ;+ ƒ l ;,,, <=>=? ######################@,+HD l and l = l. Using [ / = D / for 0 0, we obtain the coefficients of fundamental unit for = ( )%* ). A = HD l +D l +D and B = D l We can obtain following conclusions from Main Theorem. Remark 3.8. We can say similar things of Remark 3. for Main Theorem. Corollary 3.9. Let d be a square free positive integer congruent to modulo 4. If we assume that is satisfying the conditions in Main Theorem, then it always holds Yokoi s invariant =0. Proof. Yokoi s invariant is defined = * ) 3 the case of () if we substitue A and B into the 4 by Yokoi (99-993). In ( ), then we obtain D l = B = A HD l +D l +D So, we get =0 since A > B for H > 0 positive integer. In a similar way, we obtain = * ) 3 4 = 5 ( ) reason (A > B for H > 0 ) in the case of (). l 3 ƒ l 3 % l % lwx 6 =0 with same Corollary 3.0. Let be the square free positive integer corresponding to Q holding () in the Main Theorem. We state the following Table 6 where fundamental unit is, integral basis element is and Yokoi s invariant is for H =, and l. (In this table, we rule out l = 6 for H = and l = for H = since d is not a square free positive integer.) Malaysian Journal of Mathematical Sciences 3

18 Özen Özer Table 6: Square-free positive integers d with l. Œ H Œ Œ Œ Œ 5 L;,3 ####M ;,, #######$ L4;,, #######M L6;,,, ############M L4;,,,7 ############M L;,,,43 ############M L35;,,,69 #############M L56;,,, ##############M 3485 L90;,,, ##############M L5;,,9 ######M L7;,,,3 ##########M L;,,, ############M L7;,,,33 ############M ;,,,53 ############$ + 37f f f f f f f f 0+ 0f f f f f 4 Malaysian Journal of Mathematical Sciences

19 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields Table 6 (continued) : Square-free positive integers d with l ΠH ΠΠΠΠ;,,,85 ############$ f L69;,,,37 ##############M L;,,, ##############M ;,,,357 ##############$ f f f Proof. This Corollary is obtained from main theorem by taking H = or in the case of () of Main Theorem. We know is defined = 5 ( ) * ) 36. If we substitue A and B into the, then we get = A B = HD l +D l +D = + D l+d D l D l for H =. For l =, we get = 5 as well as = 4 for l = 3. Since D l is monotone increasing sequence, we obtain,78 > HD l +D l +D D l } > for l 4. Also, in the case of H =, we get = 5 for l = 3 besides = 4 for l 3 by using similar way. The proof of Corollary is completed. Corollary 3.. Let be the square free positive integer positive integer corresponding to Q holding () in the Main Theorem. We state the following Table 7 where fundamental unit is, integral basis element is and Yokoi s invariant is for H =,3 and 3 l. Malaysian Journal of Mathematical Sciences 5

20 Özen Özer Table 7 : Square-free positive integers d with 3 l. Œ Œ Œ Œ Œ 7 3 L;,,3 ######M 05 6 L5;,,,9 ##########M 8+ 7f f ;,,,35 ############$ f ;,,,45 ##############$ f L4;,,7 ######M 6+ 65f ;, ###########$, f L5;,,,03 ##############M f L7;,,,433 ##############M f Proof. By subsituting H = or 3 into the () of Main Theorem, we get this corollary and the table in the case of () of Main Theorem. If we substitue A and B into the = 5 ( ) * ) 36. then we get = A B = HD l +D l +D = + D l+d D l D l for H =. We obtain = for l = 3 Since D l is increasing sequence, we obtain,8 > HD l +D l +D D l } > for l > 3. Also, if we take H = 3, we get =4 for l = 3 as well as = 3 for l > 3 in a similar way. 6 Malaysian Journal of Mathematical Sciences

21 Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields 4. Conclusion Quadratic fields have applications in different areas of mathematics such as quadratic forms, algebraic geometry, discrete mathematics, diophantine equations, algebraic number theory, computer science and even cryptography. In this paper, we are interested in the concept of real quadratic number fields. We considered the continued fraction expansions, fundamental unit and Yokoi invariants in the terms of Fibonacci sequence. Also, we established general important and interesting results for that. Results obtained in this paper provide us a practical method so as to rapidly determine continued fraction expansion of, fundamental unit and Yokoi invariants, for period length l. We are sure that these results help the researchers to enhance and promote their studies on quadratic fields to carry out a general framework for their applications in life. References Benamar, H., Chandoul, A. and Mkaouar, M. (05). On the Continued Fraction Expansion of Fixed Period in Finite Fields. Canad. Math. Bull. 58: Friesen, C. (988 ). On continued fraction of given period. Proc. Amer. Math. Soc., 03: 9 4. Halter-Koch, F. (99). Continued fractions of given symmetric period. Fibonacci Quart., 9(4): Kawamoto, F. and Tomita, K. (008). Continued fraction and certain real quadratic fields of minimal type. J.Math.Soc. Japan. 60: Louboutin, S. (988). Continued Fraction and Real Quadratic Fields. J. Number Theory. 30: Mollin, R. A. (996). Quadratics. Boca Rato, FL: CRC Press. Olds, C. D. (963). Continued Functions. New York: Random House. Perron, O. (950). Die Lehre von den Kettenbrüchen. New York, Chelsea, Reprint from Teubner Leipzig. Sasaki, R. (986). A characterization of certain real quadratic fields. Proc. Japan Acad. Ser. A. 6 (3): Sierpinski, W. (964). Elementary Theory of Numbers. Warsaw: Monografi Matematyczne. Malaysian Journal of Mathematical Sciences 7

22 Özen Özer Tomita, K. (995). Explicit representation of fundamental units of some quadratic fields. Proc. Japan Acad. Ser. A.7 (): Tomita, K. and Yamamuro, K. (00). Lower bounds for fundamental units of real quadratic fields. Nagoya Math. J. 66: Williams, K. S. and Buck, N. (994). Comparison of the lengths of the continued fractions of D and + D. Proc. Amer. Math. Soc. 0(4): Yokoi, H. (990). The fundamental unit and class number one problem of real quadratic fields with prime discriminant. Nagoya Math. J. 0: Yokoi, H. (99). The fundamental unit and bounds for class numbers of real quadratic fields. Nagoya Math. J. 4: Yokoi, H. (993). A note on class number one problem for real quadratic fields. Proc. Japan Acad. Ser. A. 69: -6. Yokoi, H. (993). New invariants and class number problem in real quadratic fields. Nagoya Math. J. 3: Zhang, Z. and Yue, Q. (04). Fundamental units of real quadratic fields of odd class number. Journal of Number Theory. 37: 9. 8 Malaysian Journal of Mathematical Sciences