International Research Journal of Engineering and Technology (IRJET) e-issn: ON THE BINARY QUADRATIC DIOPHANTINE EQUATION

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1 Iteratioal Research Joural of Egieerig ad Techolog (IRJET) e-issn: Volume: 0 Issue: 04 Jul-0 p-issn: ON THE BINARY QUADRATIC DIOPHANTINE EQUATION = 0 S.Vidhalakshmi, M.A.Gopala, S.Nadhii, Professor, Departmet of Mathematics, SIGC, Trich-6000, Tamiladu, vidhasigc@gmail.com, mailgopala@gmail.com, M.Phil Scholar, Departmet of Mathematics, SIGC,Trich-6000,Tamiladu adhiisampath4@gmail.com *** Abstract: The biar quadratic equatio The Diophatie equatio represetig - + hperbola. 8 = 0 represets a I this paper we obtai a sequece of its itegral solutios ad preset a few iterestig relatios amog them. Ke Words: Biar quadratic equatio, Itegral solutios. MSC subject classificatio: D09.. INTRODUCTION: The biar quadratic Diophatie equatios (both homogeeous ad o homogeeous) are rich i variet 6. I7 6 the biar quadratic o-homogeeous 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 = 0. The recurrece relatios satisfied b the solutios ad are give. Also a few iterestig properties amog the solutios are ehibited.. METHOD OF ANALYSIS: the biar quadratic equatio to be solved for its o-zero distict itegral solutio is = 0 () 0, IRJET.NET- All Rights Reserved Page 8 Note that () is satisfied b the followig ozero iteger pairs (8,8), (8,6), (-8,-4), (6,7), (-4,-6). However, we have solutios for (), which are illustrated below: Solvig () for, we ve - [( -8) 08 4] () Let 08 4 which is writte as ( -4) 6 Y = 6 () Y = 4 (4) The least positive iteger solutio of () is

2 Iteratioal Research Joural of Egieerig ad Techolog (IRJET) e-issn: Volume: 0 Issue: 04 Jul-0 p-issn: , Y = Now, to fid the other solutio of (), cosider the Pellia equatio Y = () whose fudametal solutio is ~ ~, Y ) (4,9) ( 0 0 The other solutios of () ca be derived from the relatios f ~ ~ Y g (Y 4) (9) The recurrece relatios satisfied b, are respectivel s A few umerical eamples are preseted i the table below. f [(9 4 ) (9 4 ) ] g [(9 4 ) (9 4 ) ], =0,,4, Applig the lemma of Brahmagupta betwee ~ (, Y ) & ( ~ 0 0, Y ), the other solutios of () ca be obtaied from the relatios 6g 7 f Y 6 f 7 g (6) (7) Takig positive sig o the R.H.S of () ad usig (4), (6) & (7), the o-zero distict iteger solutios of the hperbola () are obtaied as follows, (Y 8 ) (8) A few iterestig relatios amog the solutios are preseted below. ) & are alwas eve ) ) 0(mod ) (6) 0(mod ) 8 is a ast umber. 4) is a quadratic iteger. ) , IRJET.NET- All Rights Reserved Page 86

3 Iteratioal Research Joural of Egieerig ad Techolog (IRJET) e-issn: Volume: 0 Issue: 04 Jul-0 p-issn: ) 7) 8) 9) is a cubic iteger. [ { ( )} ] ) Remarkable observatios: ) B cosiderig suitable liear trasformatios betwee the solutios of (), oe ma get iteger solutios for the hperbola. 0U V 4880 U ( ) 6 0) ) 46 ) ) ) 44 ) V U V U ( V ( ) 780) ) B cosiderig suitable liear trasformatios betwee the solutios of (), oe ma get iteger solutios for the parabola. N 6) M ) M N M 0760 N , IRJET.NET- All Rights Reserved Page 87

4 Iteratioal Research Joural of Egieerig ad Techolog (IRJET) e-issn: Volume: 0 Issue: 04 Jul-0 p-issn: M N []. Dickso. L. E., (9) Histor of The Theor of Numbers, Vol.II, Chelsia Publicatig Co, New York. [4]. Mordell,L,J., (969) Diophatie Equatios, Acadamic Press,Lodo. 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., (99) A Moder Itroductio to Aciet Idia Mathematics,Wile Easter Limited, Lodo. []. Carmichael, R.D., (90)The Theor of Numbers ad Diophatie Aalsis, Dover Publicatios, New York. []. Nigel,P.Smart., (999) TheAlgorithm Resolutios of Diaphatie eqatios,cambridge Uiversit, Press, Lodo. [6]. Telag,S.G., (996) Number theor, Tata Mc Graw-Hill Publishig Compa, NewDelhi. [7]. Gopala,M.A., ad Parvath,G., (00) Itegral Poits O The Hperbola , Atarctica J.Math,Vol (),49-. [8]. Gopala,M.A, Vidhalakahmi,S, Sumathi.G ad Lakshmi.K, Sep (00) Itegral Piots O The Hperbola Bessel J.Math. Vol (),9-64. [9]. Gopala,M.A.,Gokila,K.,adVidhalakahmi,S., (007) O the Diophatie Equatio 4 6 0, ActaCiecia Idica, Vol.XXXIIIM No, p , IRJET.NET- All Rights Reserved Page 88

5 Iteratioal Research Joural of Egieerig ad Techolog (IRJET) e-issn: Volume: 0 Issue: 04 Jul-0 p-issn: [0]. Telag,S.G.,(996), Number theor, Tata Mc Graw-Hill Publishig Compa, NewDelhi [].Gopala,M.A.,Vidhalakahmi,S.,ad Devibala,S., (007)O The Diophatie Equatio 4,Acta CieciaIdica,Vol.XXXIII M.No,P [].Gopala.M.A., ad Jaaki.G., (008) Observatios o, ImpactJ.Sci.,Tech, Vol()p.4, -48. [].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. [] Mollio,R.A, (998) All Solutios of the Diophatie Equatios X DY Far EastJ,Math.Sci., Speical Volume,Part III,p.7-9. [6].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. [7].Vidhalakahmi. S, Gopala,M.A ad Lakshmi.K, (August 04) Iteger Solutio of the Biar Quadratic quatio 0, Iteratioal Joural of Iovative Sciece Egieerig &Techolog, Vol.(6), [4]. Gopala,M.A, Vidhalakahmi,S, Lakshmi.K ad Sumathi.G, (0) Observatio o , Diophatus J.Maths.Vol.(),-. 0, IRJET.NET- All Rights Reserved Page 89