ON THE BINARY QUADRATIC DIOPHANTINE EQUATION

Transcription

1 Iteratioal Joural of Scietific Egieerig ad Applied Sciece (IJSEAS) - Volume-, Issue-4, Jue 0 ISSN: ON THE BINARY QUADRATIC DIOPHANTINE EQUATION M.A.Gopala * V.Geetha, D.Priaka. Professor, Departmet of Mathematics, SIGC, Trich-6000, Tamiladu, ;mailgopala@gmail.com, Departmet of Mathematics, Cauver College for Wome, Trich-6008, Tamiladu geetharamaa6@ahoo.i M.Phil Scholar, Departmet of Mathematics,SIGC,Trich-6000,Tamiladu ; durai.priaka@gmail.com Abstract The biar quadratic equatio represets a hperbola. I this paper we obtai a sequece of its itegral solutios ad preset a few iterestig relatios amog them. Kewords: Biar quadratic equatio, Itegral solutios. MSC subject classificatio: D09.. Itroductio The biar quadratic Diophatie equatios (both homogeeous ad o homogeeous) are rich i variet [ 6]. I [ 7 6] the biar quadratic ohomogeeous equatios represetig hperbolas respectivel are studied for their o-zero itegral solutios. These results have motivated us to search for ifiitel ma o-zero itegral solutios of a aother iterestig biar quadratic equatio give b The recurrece relatios satisfied b the solutios ad are give. Also a few iterestig properties amog the solutios are ehibited. METHOD OF ANALYSIS: The Diophatie equatio represetig the biar quadratic equatio to be solved for its o-zero distict itegral solutio is () Note that () is satisfied b the followig o-zero distict iteger pairs However, we have the solutios for (), which is illustrated below: solvig () for, we ve [ -6 ± 96 6] () Let 96 6 α which is writte as, ( - 48) Y α () α 04 Y 48 (4) the least positive iteger solutio of () is α 8 0, Y 88 0 Now to fid the other solutio of (), Cosider the Pellia equatio Y α ( whose fudametal solutio is ~ ~ α, Y ) (4,9) ( 0 0 The other solutios of ( ca be derived from the relatios f Y ~ α g ~ 6

2 f g [(9 4 [(9 4 Iteratioal Joural of Scietific Egieerig ad Applied Sciece (IJSEAS) - Volume-, Issue-4, Jue 0 ISSN: (9 4 (9 4 ] ], ,,4,.. Applig the lemma of Brahmagupta betwee ~ ~ (, 0 0 ) & ( s, s ) A few iterestig relatios amog the solutios are preseted below: The other solutios of () ca be obtaied from the [] & are alwas eve relatios 44g [] 0(mod ) α 64 f (6) Y 44 f 64 g [] [ ] 8 is (7) a ast umber. Takig positive sig o the R.H.S of () ad usig (4), (6) & (7) the o-zero distict iteger solutio of the [4] [ ] 6 is hperbola () are obtaied as follows, a quadratic iteger. ( 6 α ) (8) [ ] 6 [] ( 48) ( ,0,,4 (9) is a cubic iteger. The recurrece relatios satisfied b, are respectivel [ ] [6] ( [0 40 7] 4 A few umerical eamples are preseted i the table below. [7] [8] [9]

3 Iteratioal Joural of Scietific Egieerig ad Applied Sciece (IJSEAS) - Volume-, Issue-4, Jue 0 ISSN: [0] U V 048 [] 07 [] [] U ( V ( U V 7 6) [4] U ( [] V [6] U V 7680 [7] [8] ( V ( 60 8 U 60) [9] [0] 6 [ ] ( ) [ ] 6 ( ( ) Remarkable observatios: )B cosiderig suitable liear trasformatios betwee the solutios of (), oe ma get iteger solutios for the hperbola.. B cosiderig suitable liear trasformatios betwee the solutios of (), oe ma get iteger solutios for the parabola. N 0M N [ ] M N 0M M [ ] 8

4 Iteratioal Joural of Scietific Egieerig ad Applied Sciece (IJSEAS) - Volume-, Issue-4, Jue 0 ISSN: N CONCLUSION : I this paper, we have made a attempt to obtai a complete set of o-trivial distict solutios for the o-homogeeous biar quadratic equatio. To coclude, oe ma search for other choices of solutios to the cosidered biar equatio ad further, quadratic equatios with multi-variables. Ackowledgemet: The fiacial support from the UGC, New Delhi (F.MRP-/4 (SERO/UGC) dated March 04) for a part of this work is gratefull ackowledged. Refereces []. Baumath.T.S., (99A Moder Itroductio to Aciet Idia Mathematics,Wile Easter Limited, Lodo. []. Carmichael, R.D., (90)The Theor of Numbers ad Diophatie Aalsis, Dover ublicatios, New York. []. Dickso. L. E., (9Histor of The Theor of Numbers, Vol.II, Chelsia Publicatig Co, New York. [4]. Gopala,M.A., ad Parvath,G., (00) Itegral Poits O The Hperbola , Atarctica J.Math,Vol (),49-. []. Gopala,M.A, Vidhalakahmi,S, Sumathi.G ad Lakshmi.K, Sep (00) Itegral Piots O The Hperbola , [6]. Gopala,M.A., Gokila,K.,ad Vidhalakahmi,S., (007), O the Diophatie Equatio 4 6 0, ActaCieciaIdica, Vol.XXXIIIM No, p [7]. Gopala,M.A.,Vidhalakahmi,S.,adDevibala,S., (007), O The Diophatie Equatio 4,Acta CieciaIdica,Vol.XXXIII M.No,P [8].Gopala.M.A., ad Jaaki.G., (008) Observatios o, ImpactJ.Sci.,Tech, Vol()p.4, -48. [9].Gopala.M.A., ad., Shamugaadham,P.,ad Vijaashakar,A., (008), O Biar Quadratic Equatio 8 0 0, Acta Ciecia Idica,Vol. XXXIVM. No.4,p.80-80c. [0]. Gopala,M.A, Vidhalakahmi,S, Lakshmi.K ad Sumathi.G, (0) Observatio o , Diophatus J.Maths.Vol.(),-. [] Mollio,R.A, (998) All Solutios of the Diophatie Equatios X DY EastJ,Math.Sci., Speical Volume,Part III,p.7-9. []. Mordell,L,J., (969)Diophatie Equatios, Acadamic Press,Lodo. []. Nigel,P.Smart., (999)The Algorithm Resolutios of Diaphatie eqatios,cambridge Uiversit, Press, Lodo. [4]. Telag,S.G., (996) Number theor, Tata Mc Graw-Hill Publishig Compa, NewDelhi. Bessel J.Math. Vol (),

5 Iteratioal Joural of Scietific Egieerig ad Applied Sciece (IJSEAS) - Volume-, Issue-4, Jue 0 ISSN: [].Vidhalakahmi,S, Gopala,M.A ad Lakshmi.K, (04) Observatio O The Biar Quadratic Equatio 8 6 0, Scholar Joural of Phsics, Mathematics ad Statistics, Vol.(), (Sep-Nov), 4-4. [6].Vidhalakahmi. S, Gopala,M.A ad Lakshmi.K, August( 04) Iteger Solutio of the Biar Quadratic Equatio 0, Iteratioal Joural of Iovative Sciece Egieerig &Techolog, Vol.(6),