On Homogeneous Ternary Quadratic Diophantine Equation

Transcription

1 IOSR Joural of Matheatics (IOSR-JM) e-issn:78-578, p-issn: 9-765X.Volue,Issue Ver. III (Ja.-Feb.6)PP O Hoogeeous Terar Quadratic Diophatie Equatio 5 x R.buselvi ad N.hila Departet of Matheatics,.D.M. College for Woe (utooous), Nagapattia 6, Tail Nadu, Idia. Departet of Matheatics, Thiru.Vi.Ka. Govt. rts College, Tiruvarur- 6, Tail Nadu, Idia. bstract: The terar quadratic hoogeeous equatios represetig hoogeeous coe give b 5 x is aaled for its o-ero distict iteger poits o it. Five differet patters of iteger poits satisfig the coe uder cosideratio are obtaied. few iterestig relatios betwee the solutios ad special uber patters ael Polgoal uber, Praidal uber, Octahedral uber, Proic uber, Stella Octagular uber ad Oblog uber are preseted. lso kowig a iteger solutio satisfig the give coe, three triples of itegers geerated fro the give solutio are exhibited. Kewords: Terar hoogeeous quadratic, itegral solutios. I. Itroductio The Terar quadratic Diophatie equatios offer a uliited field for research because of their variet [-5].For a extesive review of various probles oe a refer [6-]. This couicatio cocers with et aother iterestig terar quadratic equatio = 5x + for deteriig its ifiitel a oero itegral solutios. lso a few iterestig relatios aog the solutios have bee preseted. NOTTIONS USED T, -Polgoal uber of rak with sie. P - Praidal uber of rak with sie. Pr - Proic uber of rak. SO -Stella Octagular uber of rak. Obl -Oblog uber of rak. OH - Octahedral uber of rak. Tet -Tetrahedral uber of rak. PP - Petagoal Praidal uber of rak II. Method Of alsis The terar quadratic equatio uder cosideratio is 5 x Differet patters of solutios of () are illustrated below. Patter-I Cosider () as 5 x ssue Write as a 5 b 7 i 5 7 i 5 8 Substitutig () ad (4) i () ad eploig the ethod of factoriatio, defie i 5 x 7 i 5 a i 5 b 8 Equatig the real ad iagiar parts i the above equatio, we get DOI:.979/ Page () () () (4)

2 O Hoogeeous Terar Quadratic Diophatie Equatio 5 x x Replacig a b 8 solutios of () are give b x 7 ab a 5 b 8 7 a 5 b ab 8 8,b b i the above equatio correspodig iteger For siplicit ad clear uderstadig, takig = i the above equatios, the correspodig iteger solutios of() are give b x , 7 x, 4 Pr ) x,7 4,7 4,7 4 4 ) t 84 CP od 7 58, 8 x,, 8 Obl 896 SO od 8 ) x, 44 Pr t od 4) 74, 5 x 5),,, t 56 P od 99 84,, 6 56 od 79 6) t t, 48, Patter-II It is worth to ote that i () a also be represeted as 5 4 i i Followig the aalsis preseted above, the correspodig iteger solutio to () are foud to be x For the sake of siplicit, takig = i the above equatios, the correspodig iteger solutios of () are give b x ) x 7 4, 4 7 4, 454 CP ) x,,, 7 Pr 4 OH od 7 ) 9 x, 4, 4765 P t od , 4) x, 96 Obl t 4 od 4, 4 DOI:.979/ Page

3 O Hoogeeous Terar Quadratic Diophatie Equatio 5 x 5) x, x, 4 Pr od 5 Patter-III The geeral solutio of the give equatio is x x,, 5, 5 ) x,,, 4 PP Pr t od 6, ) x,,, 8 t Obl od, ), 6 Perfect Square 4) x, 4 Perfect Square 5),, 4 Pr t od 5 6, 6) x,,, Obl od 8 Patter- IV Equatio () ca be writte as 5 x d we get 5 x x (5) CSE : Equatio (5) ca be writte as 5 x x Fro equatio (6),we get two equatios x 5 x We get the iteger solutios are, x x, 5, 5,, 44 Pr 5 Obl od 6 ) x,,, Tet t od 5 ) 4,,, 6 Pr 7 Obl od ), Pr t od 64 4) 7, 5) For all the values of ad, x++ is divisible b CSE : Equatio (5) ca be writte as 5 x x Fro equatio (7), we have two equatios DOI:.979/ Page (7) (6)

4 O Hoogeeous Terar Quadratic Diophatie Equatio 5 x x 5 x Solve the above two equatios, we get the iteger solutios are x x,,, 5 5 x,,, SO Pr od ) x,,, 4 PP Obl t od 7 ) 4, x, 4 a perfect Square ) 4) Each of the followig expressios represets a Nast uber,,, 6x,. Geeratio of iteger solutios Let (x,, ) be a give iteger solutio of (). The, each of the followig triples of itegers satisfies (): Triple : (x,, ) x x Triple :(x,, ) x = { [ 5 () ) x + [ ) } = x = { [ 5 () ) x + [ ) } Triple : (x,, ) x x x III. Coclusio I this paper, we have preseted four differet patters of ifiitel a o-ero distict iteger solutios of the hoogeeous coe give b. To coclude, oe a search for other patters of solutio ad their correspodig properties. DOI:.979/ Page

5 O Hoogeeous Terar Quadratic Diophatie Equatio 5 x Refereces []. L.E. Dickso, Histor of Theor of ubers, Vol., Chelsea Publishig Copa, New York, 95. []. L.J. Mordell, Diophatie Equatios, cadeic press, Lodo, 969. []. dre Weil, Nuber Theor: approach through histor: fro haurapi to legedre / dre weil: osto (irkahasuser bosto, 98. [4]. Nigel P. Sart, The algorithic Resolutios of Diophatie equatios, Cabridge uiversit press, 999. [5]. Sith D.E Histor of atheatics vol.i ad II, Dover publicatios, New York 95. [6]. M.. Gopala, Note o the Diophatie equatio x + ax + b = cta Ciecia Idica, Vol.XXVIM, No:,, 5-6. [7]. M.. Gopala, Note o the Diophatie equatio x + x + = cta Ciecia Idica, Vol.XXVIM, No:,, [8]. M.. Gopala,R.Gaaath ad R.Srikath o the Diophatie equatio = x +, Pure ad pplied Matheatika Scieces Vol.LII,No: -,, 5-7. [9]. M.. Gopala ad R.buselvi O Terar Quadratic Hoogeeous Diophatie equatio x + Px + =, ulleti of Pure ad pplied Scieces Vol.4E, No:, 5, []. M.. Gopala, S.Vidhalakshi ad.krishaaoorth, Itegral solutios Terar Quadraticax + b = c a + b, ulleti of Pure ad pplied Scieces Vol.4E, No:, (5), []. M.. Gopala, S.Vidhalakshi ads,devibala, Itegral solutios of ka x + + bx = 4kα, ulleti of Pure ad pplied Scieces Vol.5 E,No:,(6),4-46. []. M.. Gopala, S.Vidhalakshi ads,devibala, Itegral solutios of 7x + 8 = 9, Pure ad pplied Matheatika Scieces, Vol.LXVI, No:-, 7,8 86. []. M.. Gopala, S.Vidhalakshi, observatio o kax + b = c, cta Cieica Idica Vol.XXXIIIM, No:,7, [4]. M..Gopala, Maju soaath ad N.Vaitha, Itegral solutios of kx + x + =, cta Cieica Idica Vol.XXXIIIM, No: 4, 7, [5]. M..Gopala ad J.Kaliga Rai, Observatio o the Diophatie Equatio = Dx +, Ipact J.Sci. Tech, Vol (), No:, 8, [6]. M..Gopala ad V.Podichelvi, O Terar Quadratic Equatio x + = +, Ipact J.Sci. Tech,Vol (), No:,8, [7]. M..Gopala ad.gaa, Pthagorea triagles ad special polgoal ubers, Iteratioal Joural of Matheatical Sciece,Vol. (9) No:-,-5, Ja-Ju. [8]. M..Gopala ad.vijaasakar, Observatios o a Pthagorea Proble, cta Cieica Idica Vol.XXXVIM, No:4,57-5, [9]. M..Gopala ad V.Padichelvi, Itegral Solutios of Terar Quadratic Equatio Z X Y = 4XY, Ipact J.Sci. Tech; Vol (5), No:, -6,. []. M..Gopala ad J.Kaligarai, O Terar Quadratic Equatio X + Y = Z + 8, Ipact J.Sci. Tech, Vol (5), No:, 9-4,. DOI:.979/ Page