INTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION

Transcription

1 Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: Volume: 04 Issue: Mr p-issn: INTEGRL SOLUTIONS OF THE TERNRY CUBIC EQUTION y ) 4y y ) 97z G.Jki 1, C.Sry,* ssistt Professors, Deprtmet of Mthemtics, Cuvery College for Wome, Tiruchirpplli, Tmil Ndu, Idi bstrct:the o-homogeeous cubic equtio with three ukows represeted by the Diophtie equtio y ) 4y y ) 97 z is lyzed for its ptters of o zero itegrl solutios. few iterestig properties mog the solutios d some specil polygol umbers re preseted. Keywords: Cubic Equtio with Three Ukows, Itegrl solutios Itroductio: Mthemtics is the lguge of ptters d reltioships d is used to describe ythig tht c be qutified. Diophtie equtios hs bee mtter of iterest to vrious mthemticis. The problem of fidig ll iteger solutios of diophtie equtio with three or more vribles d equtios of degree t lest three, i geerl presets good del of difficulties. I [1-], theory of umbers were discussed. I [4,], specil Pythgore trigle problem hve bee discussed for its itegrl solutios. I [6-10], higher order equtios re cosidered for itegrl solutios. I this commuictio, the o-homogeeous cubic equtio with three ukows represeted by the equtio y ) 4y y ) 97 z is cosidered d i prticulr few iterestig reltios mog the solutios re preseted. Nottios: Obl = Oblog umber of rk. T m, = Polygol umber of rk with sides m. CS = Cetered Squre umber of rk. SO = Stell octgul umber of rk. O = Octhedrl umber of rk. Go = Gomoic umber of rk. Str = Str umber of rk. TO = Tructed octhedrl umber of rk. m P = Pyrmidl umber of rk with sides m. 017, IRJET Impct Fctor vlue:.181 ISO 9001:008 Certified Jourl Pge 66

2 Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: Volume: 04 Issue: Mr p-issn: Method of lysis: The Cubic equtio to be solved for its o-zero itegrl solutio is y ) 4y y ) 97z 1) O substitutio of the trsformtios, u v, y u v ) i 1) leds to, u 1 v 486z ) We illustrte below four differet ptters of o-zero distict iteger solutios to 1). Ptter: 1 ssume z z, b 4) where d b re o-zero itegers. d write i 19 i Substitutig 4) & i ), d usig fctoriztio method, u 1) i v u ) i v 19 i 19 i i i 6) Equtig the like terms d comprig rel d imgiry prts, we get u u, 19 8 b 7 b b v v, 7 b 7 b 9b Substitutig the bove vlues of u & v i equtio ), the correspodig iteger solutios of 1) re give by, 4 60 b 8 b b y y, b z z, b 1. y, ), ) is cubicl iteger.. y 1,1) 1,1 ) is sty umber.. 1, ) y1, ) 8 z1, ) 0T b b 8, b,1) y b,1) SOb 9T14, b,1),1) 4 z,1) O T 6, 6Go mod 6) 4. Go mod 1. y b 0Go, ) y, ) z, ) 40P 1T, 7. y 1,1) 1,1) 8z 1,1 ) is perfect squre. mod 60) mod 189) Ptter: Isted of, Write i 9 9i 7) Substitutig 7) d 4) i ) d employig the method of fctoriztio, followig the procedure preseted i ptter 1, the correspodig iteger solutios of 1) re represeted by 017, IRJET Impct Fctor vlue:.181 ISO 9001:008 Certified Jourl Pge 666

3 Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: Volume: 04 Issue: Mr p-issn: 9-007, 18 y y, 6 70 b z z, b 1. y, ), ) z, ) is cubicl iteger.. y 1,1) 1,1) z 1,1 ) is sty umber b 70b 08 b 80b y,1),1) z,1) 6P T16, T,,1) z,1) P 6T6, 1, z 1, 90SO T 71Go b 0, b 9Gob 1, y1, z1, TOb Ob 4T, b 7. y 1,1) 1,1) 6 z 1,1 ) is perfect squre. 0Go mod 0) 017, IRJET Impct Fctor vlue:.181 ISO 9001:008 Certified Jourl Pge 667 mod 8) mod14) 04Go b mod11 Ptter: Isted of, Write i 1 i 8) Substitutig 8) d 4) i ) d followig the sme procedure preseted i ptter 1, the correspodig iteger solutios of 1) re represeted by, 4 60 b 8 b b y y, b z z, b 1. y, ), ) 8 z, ) is cubicl iteger.. y 1,1) 1,1) 8 z 1,1 ) is sty umber.. y,1),1) SO 6T Go 08 b 80b 6, 4 8, 4. y,1),1) 6 T Go ) mod 647).,1) z,1) 46P T6, T8, y,1) z,1) 4O T 0, 7. y 1,1) 1,1) 4 z 1,1 ) is perfect squre. 84Go 9Go mod ) mod 1 mod1) Ptter: 4 Isted of, Write 1 9i 1 9i ) d usig the sme procedure s i ptter 1, the correspodig solutios of ) re represeted by 1 u u, 1 9 b 4 b 7b ) 1 v v, 9 4 b 9 b 6b ) z z, b

4 Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: Volume: 04 Issue: Mr p-issn: Sice our iterest is o fidig iteger solutios, we hve choose d b suitbly so tht u, v d z re itegers. Let us tke = d b = B, we hve u u, 117 7B 91 B 6B v v, 61 z z, 9 91B 4B 1 B 8B I view of ), the iteger solutios of 1) re give by, 78 v v, 44 z z, 9 670B 4B 160B B 940B B 7110B 1. y, ), ) 40 z, ) is cubicl iteger.. y 1,1) 1,1 ) is sty umber.. 4..,1) y,1) 468P 688T8, 444Go mod 860,1) 4 z,1) 4T 400Go mod 9) y 4,,1) 6 z,1) 474T0, 46Go,1) y,1) z,1) 468P 894T0, 1,1) y 1,1) 7 z 1,1 is perfect squre. 7. ) mod 66 1Go Note: I dditio, oe my write 486 s 1 9i 1 9i 117 4i 117 4i 11 47i 11 47i i 87 7i 7 67i 7 67i 69i 69i For these choices, oe my obti differet ptters of solutios of 1). mod 7807) 017, IRJET Impct Fctor vlue:.181 ISO 9001:008 Certified Jourl Pge 668

5 Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: Volume: 04 Issue: Mr p-issn: Coclusio: I this pper, we hve preseted four differet ptters of o-zero distict iteger solutios of the ohomogeeous coe give by y ) 4y y ) 97 z. To coclude, oe my serch for other ptters of o-zero iteger distict solutios d their correspodig properties for other choices of cubic diophtie equtios. Refereces: [1] Crmichel, R.D., The theory of umbers d Diophtie lysis, Dover Publictios,New York, 199. [] Dickso L.E, History of Theory of Numbers, Vol.11, Chelse Publishig compy, New York,19. [] Mordell. L.J, Diophtie equtios, cdemic Press,Lodo,1969 Telg, S.G., Number theory, Tt Mc Grw Hill publishig compy, New Delhi, 1996 [4] Gopl. M., Mju Somth d Vith. N, Prmetric Solutios of XXXIII,, , 007. [] Gopl.M. d Jki.G, Itegrl solutios of J.Sci.,Tech., 41), 97-10, y z, ct cieci idic, y ) y y z w ) p, Impct [6] Gopl M.., Pdichelvi.V, observtios o the terry cubic equtio y y 4 z z ) rchimedes J.Mth 11), 1-7, 011. [7] Gopl M., Srividhy.G., Itegrl solutios of terry cubic diophtie equtio y z ct Cieci Idic,Vol XXXVII, No.4, , 011. [8] Gopl.M.., d Jki.G., Itegrl solutios of y ) y y) z w ) p, Impct jourl of Sciece & Techology, Vol-4, No.97-10, 010. [9] Jki.G d Sry.C., Observtios o the Terry Qudrtic Diophtie Equtio 6 z y ) 1y y 9 7, Itertiol Jourl of Iovtive Reserch i Sciece, Egieerig d Techology, Vol-, Issue-, Pg.o: , Feb 01 [10] Jki.G d Sry.P., O the terry Cubic diophtie equtio z y ) 6y 4 y) 4 40, Itertiol Jourl of Sciece d Reserch- olie, Vol, Issue, Pg.No:7-9, Mrch , IRJET Impct Fctor vlue:.181 ISO 9001:008 Certified Jourl Pge 669