Observations on the Non-homogeneous Quintic Equation with Four Unknowns

Transcription

1 Itertiol Jourl of Mthemtic Reerch. ISSN Volume, Number 1 (13), pp Itertiol Reerch Publictio Houe Obervtio o the No-homogeeou Quitic Equtio with Four Ukow S. Vidhylkhmi 1, K. Lkhmi d M.A. Gopl 3 Deprtmet of Mthemtic, Shrimti Idir Gdhi College, Trichy 6, Idi E-mil: 1 vidhyigc@gmil.com, lkhmi1664@gmil.com, 3 myilgopl@gmil.com Abtrct We obti ifiitely my o-zero iteger qudruple ( x, y, z, w ) tifyig the quitic equtio with four ukow. xy 6 z ( k ) w.vriou iteretig reltio betwee the olutio d pecil umber, mely, polygol umber, pyrmidl umber, Jcobthl umber, Jcobthl-Luc umber, keye umber, Four Dimeiol Figurtive umber d Five Dimeiol Figurtive umber re exhibited. Keyword: Quitic equtio with four ukow, itegrl olutio, - dimetiol, 3-dimetiol, 4-dimetiol d -dimeiol Figurtive umber. MSC Mthemtic ubject clifictio: 11D41. Nottio ( 1)( m ) Tm, 1 -Polygol umber of rk with ize m m 1 P ( 1)(( m ) m -Pyrmidl umber of rk with 6 ize m SO ( 1) -Stell octgulr umber of rk S 6 ( 1) 1-Str umber of rk

2 18 S. Vidhylkhmi et. l. PR ( 1) -Proic umber of rk 1 ( 1) J -Jcobthl umber of rk 3 j ( 1) -Jcobthl-Luc umber of rk KY ( 1) -keye umber. ( 1)( )( 3)( 4) F,,3 = Five Dimeiol Figurtive! umber of rk whoe geertig polygo i trigle. ( 1)( )( 3) F4,,3 = Four Dimeiol Figurtive umber of 4! rk whoe geertig polygo i trigle Itroductio The theory of diophtie equtio offer rich vriety of fcitig problem. I prticulr,quitic equtio, homogeeou d o-homogeeou hve roued the iteret of umerou mthemtici ice tiquity[1-3]. For illutrtio, oe my refer [4-1] for quitic equtio with three,four d five ukow. Thi pper cocer with the problem of determiig o-trivil itegrl olutio of the ohomogeeou quitic equtio with four ukow give by xy 6 z ( k ) w.a few reltio betwee the olutio d the pecil umber re preeted. Method of Alyi The Diophtie equtio repreetig the quitic equtio with four ukow uder coidertio i xy 6 z ( k ) w (1) Itroductio of the trformtio x u v, y u v, z v () i (1) led to u v ( k ) w (3) The bove equtio (3) i olved through three differet pproche d thu, oe obti three ditict et of olutio to (1) Approch 1: Let w b (4) Subtitutig (4) i (3) d uig the method of fctoritio, defie ( u i v) ( k i ) ( i b) r exp( i )( i b)

3 Obervtio o the No-homogeeou Quitic Equtio with Four Ukow 19 by where r k, t () k Equtig rel d imgiry prt i () we get u r [co ( b 1 b ) i ( b b b )] 1 i v r b b b b b I view of () d (4), the correpodig vlue of [co ( )] ( 1 )] x r b b b b b [co ( 1 ) i ( b 1b ( b b b )] y r b b b b b [co ( 1 ) x, y, z d w re repreeted i ( b 1b ( b b b )] 4 3 i 3 4 z r [co ( b b b ) ( b 1 b ) w b (6) Propertie 1. x(, b) y(, b) z(, b) 76i 3. z(, ) r (co )(4 T3, 1. P 6P 6 T3, PR T4, ) 3. ( w(,1) 3 J 4) i ty umber. 4. ( w(,1) j4 ) i cubicl iteger Remrk 1 To lye the ture of the olutio, oe h to go for prticulr vlue to. For implicity d cler udertdig, tkig i (6), the correpodig vlue of the iteger qudruple ( x, y, z, w ) tifyig xy 6z w re repreeted by x b 1b b b b y b b b b b z b b b w b The iteger qudruple ( x, y, z, w ) repreeted by (7) tifie the followig propertie: (7)

4 13 S. Vidhylkhmi et. l. x (, b ) y (, b ) 1 T T (mod) 1. 4, 3, 3. 4F 48F 3P 4T 34T 1 T x(,1) y(,1),,3 4,,3 3, 4, 6, [ z(, ) y(, ) x(, ) 4F 9P 7 T ] i ty umber.,,3 3, 4[ y (1, b ) z (1, b ) w (1, b ) T 8 T T 17 T ] i cubicl 4. 3, b 6, b 4, b, b iteger. 8[ x (1, b ) y (1, b ) T 3 T ] i biqudrtic iteger.. 3, b 4, b 6. ( z,1) w(,1) 66 j4 1 j 7. x(,1) y(,1) z(,1) 9 1KY 4J 8. If ( x, y, z, w ) i y give olutio of (1) the ech of the followig qudruple tifie (1): ( i)(6x y 6 z, x 6y 1 z, x y 11 z, w ) ( ii)( 3x 8y 4 z,x 3y 1 z, x y 7 z, w ) ( iii)( 6x 49y 84 z, x 6y 1 z, x 7y 13 z, w ) ( iv)( x 3y 1 z,3x y 1 z, x y z, w ) ( v)(3x y 1 z, x 3y 1 z, x y z, w ) Remrk It i lo worth metioig here tht we get iteger olutio whe 1,,3,... i (1). For the ke of udertdig we exhibit below the itegrl olutio correpodig to 1, k 1,, k 1, Exhibit 1: 1, k 1 x 1 b b b b 1b y 3 b 3 b 1b 4 3 z b b b b 1b w b Exhibit : 1, k x 3 1 b 37b 1 b 1 b 7b y b 1b 3 b 3 b 17b z 1 b 1 b b b 1b w b

5 Obervtio o the No-homogeeou Quitic Equtio with Four Ukow 131 Exhibit 3:, k 1 x 1 b b 7 b 7 b 3b y b b b b b ( 6)[ 1 ] z b b 1b 1 b b w b Exhibit 4:, k x 3 1 b 37b 1 b 1 b b y b b b b b ( 1 ) 19( ) z b b b 4 b b w b Approch Tkig i (3), we hve, u v w (8) whoe olutio i give by 3 4 u b 1b 4 3 v b b b Agi tkig 1 i (3), we hve, u v ( k ) w (9) whoe olutio i repreeted by u ku v 1 v1 u ku The geerl form of itegrl olutio to (8) i give by A i B u u i v B v A, 1,,3,... where ( k i) ( k i) A ( k i) ( k i) B Thu i view of (), the followig qudruple of iteger bed o ( x, y, z, w ) lo tify (1) Qudruple: ( x, y, z, w ),where

6 13 S. Vidhylkhmi et. l. x u v A i v u B y u v A i v u B u z v A i ( ) B w b ( ) ( ) ( ) ( ) The bove vlue of x, y, z tify the followig recurrece reltio repectively x kx ( k ) x 1 y ky k y 1 ( ) z kz k z 1 ( ) Propertie 1 3 x(,, k) y(,, k) (1 A 4i B )[1F,,3 4F4,,3 1P 3T3, T4, T6, ] 1. S 3PR 1T 6SO 6T 3T 6 w(, b) 1 b 4, 3, b 4, b. i 3. x(,, k) y(,, k) (4A 1 B )[8 T3, 1. Pp SO P R T4, ] ib z (,1, k) A (1F 4,,3 4( OH ) 9T 4, 3T 3, ) (4T 3, 1. P 94P 76T 4, 3 PR ) i. z(,, k) (A 76 B )(8 T3, 1. P SO T3, T4, ) Approch 3 Subtitutig (4) i (3) d uig the method of fctoritio, defie ( u i v) ( k i ) ( i b) Expdig biomilly d equtig rel d imgiry prt, we hve u f k b b g k b b ( )( 1 ) ( )( ) v g k b b f k b b where r r r f ( x) ( 1) mcrk r 1 r1 r1 r1 g( x) ( 1) c r1k r ( )( 1 ) ( )( ) (1)

7 Obervtio o the No-homogeeou Quitic Equtio with Four Ukow 133 I view of () d (1) the correpodig iteger olutio ( x, y, z, w ) to (1) i obtied x [ f ( k) g( k)]( b 1 b ) [ f ( k) g( k)]( b b ) y f k g k b b f k g k b b [ ( ) ( )]( 1 ) [ ( ) ( )]( ) z g k b b f k b b ( )( 1 ) ( )( ) w b Cocluio I cocluio, oe my get differet ptter of olutio to (1) d their correpodig propertie. Referece [1] L.E. Dicko,Hitory of Theory of Number,Vol.11,Chele Publihig compy, New York (19). [] L.J. Mordell, Diophtie equtio, Acdemic Pre, Lodo(1969). [3] Crmichel,R.D.,The theory of umber d Diophtie Alyi,Dover Publictio, New York (199) [4] M.A. Gopl & A.Vijyhkr, A Iteretig Diophtie problem 3 3 x y z, Nro Publihig Houe, Pp 1-6, 1. [] M.A. Gopl & A.Vijyhkr, Itegrl olutio of terry quitic Diophtie equtio x (k 1) y z,itertiol Jourl of Mthemticl Sciece 19(1-), ,(j-jue 1) [6] M.A. Gopl,G.Sumthi & S.Vidhylkhmi, Itegrl olutio of ohomogeeou terry quitic equtio i term of pell equece 3 3 x y xy( x y) Z,ccepted for Publictio i JAMS(Reerch Idi Publictio) [7] S. Vidhylkhmi, K.Lkhmi d M.A.Gopl, Obervtio o the homogeeou quitic equtio with four ukow x y z ( x y)( x y ) w,ccepted for Publictio i Itertiol Jourl of Multidicipliry Reerch Acdemy(IJMRA) [8] M.A. Gopl & A.Vijyhkr, Itegrl olutio of o-homogeeou quitic equtio with five ukow xy zw R, Beel J.Mth.,1(1),3-3,11. [9] M.A. Gopl & A.Vijyhkr, olutio of quitic equtio with five ukow x y ( z w ) P,ccepted for Publictio i Itertiol Review of Pure d Applied Mthemtic. [1] M.A. Gopl, G. Sumthi & S.Vidhylkhmi, O the o-homogeou quitic equtio with five ukow x y z w 6T,ccepted for Publictio i Itertiol Jourl of Multidicipliry Reerch Acdemy (IJMRA).