Ascertainment of The Certain Fundamental Units in a Specific Type of Real Quadratic Fields

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1 J. Ana. Nm. Theor. 5, No., 09-3 (07) 09 Jorna of Anaysis & Nmber Theory An Internationa Jorna Ascertainment of The Certain Fnamenta Units in a Specific Type of Rea Qaratic Fies Zbair Nisar an Sajia Kosar Department of Mathematics an Statistics, Internationa Isamic University Isamaba, Pakistan Receive: 7 Mar. 07, Revise: 4 Jn. 07, Accepte: 7 Jn. 07 Pbishe onine: J. 07 Abstract: The aim of the paper is to cassify the rea qaratic nmber fies Q( ) having specific form of contine fraction expansions of agebraic integer w an is to etermine the genera expicit parametric representation of the fnamenta nit ε for sch rea qaratic nmber fies where,3(mo4) is a sqare free positive integer. Aso, Yokoi s -invariants n an m wi be cacate in the reation to contine fraction expansion of w for sch rea qaratic fies. Keywors: Contine Fraction Expansion, Fnamenta Unit, Qaratic Fie. AMS Sbject Cassification: A55, R7, R Introction Preiminaries Definition.{R i } seqence is efine by recrrence reation as foows: In Nmber Theory, rea qaratic nmber fies have great importance. Many researchers have obtaine their rests on the rea qaratic nmber fies (-). The athor (7-) consiere some types of rea qaratic fies an etermine their fnamenta nit as we as Yokoi s invariants. The prpose of this paper is to sty on a particar rea qaratic fie an etermine to cassification of rea qaratic fies incing the contine fraction expansion which has got the partia qotients are eqa to 8 in the symmetric part of perio ength by consiering (7-). Aso, Yokoi s invariants are cacate an presente in tabes. In any k = Q( ) rea qaratic nmber fie, integra basis eement of agebraic integers ring in rea qaratic fies is etermine by w = = a 0 ;a,a,...,a (),a 0 in the case of,3(mo4) where = () is the perio ength of contine fraction expansion. The fnamenta nit ε of rea qaratic nmber fie is aso enote by ε = (t + )/ >. Aso, Yokoi s invariants are t expresse by n = an m = where x represents the greatest integer ess than or eqa to x. t R i = 8R i + R i for i with initia vaes R 0 = 0 an R =. Definition.Let c n = ac n + bc n recrrence reation of {c n } seqence where a,b are rea nmbers. The poynomia is cae as a characteristic eqation if it is written in the form: x ax b= 0 For or seqence, it can be written for each eement of seqence as foows: R k = (4+ 7) k (4 7) k 7 for k Remark.Let {R n } be the seqence efine as in Definition. Then, we state the foowing: { 0(mo 4), n 0(mo); R n = (mo 4), n (mo). for n 0. Corresponing athor e-mai: from.fatehjang@gmai.com c 07 NSP

2 0 Z. Nisar, S. Kosar: Ascertainment of the certain fnamenta nits... Lemma.For a sqare-free positive integer congrent to,3 moo 4, we pt w =, a 0 = w, w R = a 0 + w. Then w / R(), bt w R R() hos. Moreover for the perio=() of w R, we get w R = a 0,a,...,a an w = a 0,a,...,a,a 0. Frthermore, et w R = (P w R + P ) Q w R + Q = a 0,a,,a,w R be a moar atomorphism of w R, then the fnamenta nit ε of Q( ) is given by the foowing forma: an ε = t + =(a 0 + )Q () + Q ( ) > t = a 0.Q () + Q (), = Q (). where Q i is etermine by Q 0 = 0, Q = an Q i+ = a i Q i + Q i,(i ). Proof.Proof is omitte in 6, Lemma Lemma.Let be the sqare free positive integer congrent to,3 moo 4. We wi consier w which has got partia constant eements repeate 8s in the case of perio =(). If we et a 0 enote the a 0 = the integer part of w for congrent to,3(mo 4), then we have contine fration expansions w = =a 0 ;a,a,...,a (),a () =a 0 ;8,8,,8,a 0 for qaratic irrationa nmbers an w R = a 0 + = a 0,8,...,8 for rece qaratic irrationa nmbers. In the contine fraction w R = a 0 + = b,b,...,b n,... = a 0,8,...,8,...,P k = a 0 R k + R k an Q k = R k are etermine in the contine fraction expansion where P k an Q k are two seqences efine by: for j 0 P = 0,P 0 =,P j+ = b j+.p j + P j, Q =,Q 0 = 0,Q j+ = b j+.q j + Q j, Proof.It can be prove easiy by consiering references (7-). 3 Main Rests Theorem.Let be a sqare free positive integer an be a positive integer satisfying that. Sppose that parameterizations of is = µ R + µ(8r +R )+7 for µ integer. If µ is o positive integer, we have, 3(mo4) an w = µr + 4;8,8,...,8,µR }{{} with =(). Moreover, we can get fnamenta nit ε, coefficients of fnamenta nit t, as foows: ε =(µr + 4)R + R + R, t = (µr + 4)R + R an µ = R. Proof.Let be the positive integer. Using Remark, we get R (mo4) an R 0(mo4) for (mo4) an 3(mo4). By consiering µ is o positive integer an sbstitting these eqaities into the = µ R + µ(8r +R )+7, we get (mo4). Aso, we have that R 0(mo4) an R (mo4) for both 0(mo4) an (mo4). By consiering µ is o positive integer an sbstitting these eqaities into the = µ R + µ(8r + R )+7, so 3(mo4) hos. On sbstitting w into the w R, we get w R =(µr + 4)+ µr + 4;8,8,...,8,µR }{{} an we have w R =(µr + 8)+... w R =(µr + 8)+...+ w R Using Lemma. an Lemma. abot the properties of contine fraction expansion, we get w R =(µr + 8)+ R w R + R R w R + R an by sing Definition into the above eqaity, we obtain, w R (µr +8)w R (+µr )=0 This reqires that w R = (µr + 4)+ since w R > 0. Aso, sing Lemma, we get w = = µr + 4;8,8,...,8,µR }{{} + 8 an = (). This competes the first part of theorem. Now, to etermine ε, t an sing Lemma, we get Q = =R,Q = a.q + Q 0 Q = 8=R Q 3 = a Q +Q = 8R +R = 65=R 3,Q 4 = 58=R 4, So, this impies that Q i = R i by sing mathematica inction i 0. If we sbstitte these vaes of the seqence into the c 07 NSP

3 J. Ana. Nm. Theor. 5, No., 09-3 (07) / ε = t + = (a 0 + )Q () + Q () > an rearranging, we wi get ε =(µr + 4)R + R + R, t = (µr + 4)R + R an = R. Coroary.If is a sqare free positive integer an is a positive integer satisfying that as we as parametrization of as foows: = R + (4R +R )+7 then, we have,3(mo4) an w = R + 4;8,8,...,8,µR }{{} + 8 with = (). Aso, fnamenta nit ε, coefficients of fnamenta nit t, an Yokoi invariant as foows: ε =(R + 4)R + R + R, t = (R + 4)R + R an = R Aso, we have vae of Yokoi s -invariant m =. Proof.This coroary is obtaine by sing Theorem with taking µ =. So, we sho etermine vae of the Yokoi s invariant m. We know that m = from H. Yokoi s references. If we sbstitte t an into the m, then we get m = 4R = t R + 8R =, +R since R increasing seqence an 4R,984< R + 8R < for. Therefore we + R obtain m = for owing to efinition of m. Besies, Tabe is given as nmerica istrates. (In this tabe, () =,3,6,7,8 are re ot since is not a sqare free positive integer in these perios). Coroary.If is a sqare free positive integer an is a positive integer satisfying that as we as parametrization of is t = 9R + 4R +6R + 7 then, we have,3(mo4) an w = 3R + 4;8,8,...,8, 6R }{{} with=(). Aitionay, we obtain fnamenta nit ε, coefficients of fnamenta nit t, as foows: ε =(3R + 4)R + R + R, t = (3R + 4)R + R an = R Aso, we have Yokoi s -invariant vae n = Proof.This coroary is obtaine by sing theorem for µ = 3. In the same manner we obtain n = for owing to efinition of n. Besies, foowing Tabe gives an exampe for this coroary. (In this tabe, we aso re ot()=3 since is not a sqare free positive integer in this perio). Theorem.Let be the sqare free positive integer an be a positive integer hoing that = 0(mo) an. We assme that parametrization of is = µ R 4 +(4R + R )µ+ 7. for µ > 0 positive integer. If µ (mo4) positive integer then (mo4) an µr w = + 4;8,8,...,8, µr }{{} + 8 hos for = (). Moreover, the foowing eqaities aso ho: µr ) ε =( + 4R +R + R for ε,t an. t = µr + 8R + R an = R Proof.Let 0(mo) an > ho. If 0(mo) hos then we have R 0(mo4),R (mo4). Consiering µ (mo4) positive integer an sbstitting these eqivaent an eqations into the parameterization of then we get (mo4). Using Lemma, we pt we get w R = µr ;8,8,...,8, µr }{{} + 8, µr w R =(µr + 8)+... w R =(µr + 8) w R Now by sing Lemma an Lemma abot the properties of contine fraction expansion, we get w R =(µr + 8)+ R w R + R R w R + R by sing inction an property of contine fraction expansion an the Definition into the above ineqaity, we obtain w R (µr + 8)w R (+ µr )=0. c 07 NSP

4 Z. Nisar, S. Kosar: Ascertainment of the certain fnamenta nits... Tabe :??? () m w ε ,8,8,8, ,8,8,8,8, ,8,8,..., ,8,8,...,8, ,8,8,..., Tabe :??? () n w ε 79 8,8, ,8,8,8,8, ,8,8,...,8, ,8,8,...,8, ,8,8,...,8, ,8,8,...,8, ,8,8,...,8, ,8,8,...,8, Tabe 3:??? () m w ε 66 8;8, ;8,8,8, ;8,8,8,8,8, ;8,8,...,8, This reqires that w R = µr + 4+ since w R > 0. Consiering Lemma, we get w = µr = + 4;8,8,,8, µr }{{} + 8 an=(). This competes the first part of theorem. Now we sho ettermine ε,t an sing Lemma, we have Q = =R,Q = a.q + Q 0 Q = 8=R, Q 3 = a Q +Q = 8R +R = 65=R 3,Q 4 = 58=R 4, This impies that Q i = R i by sing mathematica inction i 0. On sbstitting these vaes of seqence into the ε = t + =(a 0 + )Q () + Q () > an rearrange, we get for ε,t an. ε =( µr + 4R + R )+R t = R + 8R + R an µ = R Coroary 3.Let be the sqare free positive integer an be a positive integer hoing that 0(mo) an >. We assme that parameterizations of is = R 4 + 4R + R + 7 Then we get (mo4) an R + 8 w = ;8,8,...,8,R }{{} + 8 an=(). Moreover, we have foowing eqaities: R ) ε =( + 4R + R + R an t = R + 8R + R an = R {, if =; m = 3, if. Proof.We obtain this coroary by taking µ = in Theorem. In a simiar way, we get, ( 4> R ) R R > 3,938 for 4. Therefore, we obtain 4R m = R + 8R = 3 +R c 07 NSP

5 J. Ana. Nm. Theor. 5, No., 09-3 (07) / 3 for 4. Tabe 3 shows some nmerica exampes for Coroary 3. (In this tabe we re ot ()=8,0 since is not a sqare free positive integer in these perios). 4 concsion In this paper, we introce the notion of rea qaratic fie strctres sch as contine fraction expansions, fnamenta nit an Yokoi invariants in the terms of specia seqence. We estabishe a practica metho so as to rapiy etermine contine fraction of w, fnamenta nit ε an Yokoi invariants n,m for cassifie sch rea qaratic nmber fies. References C. Friesen, On contine fraction of given perio, Proc. Amer. Math. Soc., 03(), 9-4 (988). F. H. Koch, Contine fraction of given symmetric perio, Fibonacci Qart., 9(4), (99). 3 F. Kawamoto an K. Tomita, Contine fraction an certain rea qaratic fie of minima type, J. Math. Soc. Japan, 60, (008). 4 S. Lobotin, Contine Fraction an Rea Qaratic Fies; J. Nmber Theory, 30, (988). 5 R. A. Moin, Qaratics, CRC Press, Boca Rato, FL., (996). 6 C. D. Os, Contine Fnctions, New York: Ranom Hose, (963). 7 Ö. Özer, On Rea Qaratic Nmber Fies Reate with Specific Type of Contine Fractions, Jorna of Anaysis an Nmber Theory, 4(), (06). 8 Ö. Özer, Notes On Especia Contine Fraction Expansions an Rea Qaratic Nmber Fies, Kirkarei University Jorna of Engineering an Science, (), (06). 9 Ö. Özer an C. Öze, Some Rests on Specia Contine Fraction Expansions In Rea Qaratic Nmber Fies, J. of Math. Ana.7(4),98-07 (06). 0 Ö. Özer, A Note On Fnamenta Units in Some Type of Rea Qaratic Fie, AP Conference Proceeings, 773, (06); Ö. Özer, A. B. M. Saem, A Comptationa Techniqe For Determining Fnamenta Unit in Expicit Type of Rea Qaratic Nmber Fies, Internationa Jorna of Avance an Appie Sciences, 4(), -7 (07). Ö. Özer, Fibonacci Seqence an Contine Fraction Expansions in Rea Qaratic Nmber Fies, Maaysian Jorna of Mathematica Science, (), 97-8 (07). 3 O. Perron, Die Lehre von en Kettenbrehen, New York: Chesea, Reprint from Tebner,Leipzig,99., R. Sasaki, A Characterization of Certain Rea Qaratic Fies, Proc. Japan Aca., 6(3), (986). 5 W. Sierpinski, Eementary Theory of Nmbers, Warsaw: Monografi Matematyczne, J. K. Tomita, Expicit Representation of Fnamenta Units of Some Qaratic Fies, Proc. Japan Aca., 7(), 4-43 (993). 7 K. Tomita an K. Yamamro, Lower Bons for Fnamenta Units of Rea Qaratic Fies, Nagoya Math. J., 66, 9-37 (00). 8 K. S. Wiiams an N. Bck, Comparison of the engths of the contine fractions of D an (+ D), Proc. Amer. Math. Soc., 0(4), (994). 9 H. Yokoi, The fnamenta nit an cass nmber one probem of rea qaratic fies with prime iscriminant, Nagoya Math. J., 0, 5-59 (990). 0 H. Yokoi, A note on cass nmber one probem for rea qaratic fies, Proc. Japan Aca.,69, -6 (993). H. Yokoi, The fnamenta nit an bons for cass nmbers of rea qaratic fies, Nagoya Math. J., 4, 8-97(99) H. Yokoi, New invariants an cass nmber probem in rea qaratic fies, Nagoya Math. J., 3, (993). Zbair Nisar is crrenty enroe as a PhD stent of mathematics at Internationa Isamic University, Isamaba, Pakistan. He receive his MS egree at the same niversity. His fie of interest is agebra an agebraic nmber theory. Sajia Kosar is an Assistant Professor of Mathematics at Internationa Isamic University, Isamaba Pakistan. She receive the PhD egree in Mathematics at University of York, Unite Kingom. Her main research interests are agebra an agebraic nmber theory. c 07 NSP