On The Homogeneous Quintic Equation with Five Unknowns

Transcription

1 IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-78,p-ISSN: X, Volume 7, Issue 3 (Jul. - Aug. 013), PP O The Homogeeous Quitic Equtio with Five Ukows y y 3 3 ( y ) 3(( y)( z w ) P M.A. Gopl 1, S.Vidhylkshmi, S.Mllik, 3.Deprtmet of Mthemtics, Shrimti Idir Gdhi college,trichirpplli.6000 Astrct: The quitic Diophtie equtio with five ukows give y 3 3 y y ) 3(( y)( z w ) P is lyzed for its ifiitely my o-zero distict itegrl solutios. A few iterestig reltios etwee the solutios d specil umers mely, cetered polygol umers, cetered pyrmidl umers, jcosthl umers, lucs umers d kye umers re preseted. Keywords: Quitic equtio with five ukows, Itegrl solutios, cetered polygol umers, cetered pyrmidl umers. I. Itroductio The theory of Diophtie equtios offer rich vriety of fscitig prolems.i prticulr quitic equtios homogeeous or o-homogeeous hve roused the iterest of umerous mthemticis sice tiquity[1,,3].for illustrtio,oe my refer [-1],for quitic equtios with three,four d five ukows. This pper cocers with the prolem of determiig itegrl solutios of the o-homogeeous quitic 3 3 equtio with five ukows give y y y ) 3(( y)( z w ) P A few reltios. etwee the solutios d the specil umers re preseted.. Nottios t, P m m [( m ) m] 6 ( )( )( 3) m.-polygol umer of rk with size m. Pt - pettope umer of rk. ( SO 1) - Stell octgulr umer of rk. S 6( ) - Str umer of rk. Pr ( ) -Pyrmidl umer of rk with size m. - Proic umer of rk. 1 J ( ) -Jcosthl umer of rk. 3 j ( ( ) ) - Jcosthl lucs umer of rk. Ky ( ) - Kye umer. ( )( )( 3) F, m,3 -Four dimesiol figurtive umer of rk! whose geertig polygo is trigle. ( )( )( 3)( ) F, m,3 -Five dimesiol figurtive umer of! rk whose geertig polygo is trigle. m( ) CP m - Cetered polygol umer of rk with size m. 7 Pge

2 O The Homogeeous Quitic Equtio With Five Ukows II. Method of Alysis The Diophtie equtio represetig the quitic with five ukows uder cosidertio is 3 3 y y ) 3(( y)( z w ) P (1) Itroducig the trsformtios u v, y u v, z uv, w uv () i (1),it leds to u v 7 p (3) which is solved i differet wys ledig to differet solutio ptters to (1)..1 Ptter : I Assume p write 17 s 17 (1 i)(1 i) Sustitutig () d () i (3) d employig the method of fctoriztio,defie u iv ( 1 i)( i Equtig rel d imgiry prts, we get u 8 v Thus, i view of (), the o-zero distict itegrl solutios of (1) re give y, 6, z (, ( w (, ( p (, 8( 8( () () (6).1. Properties 1), ), ), ) 36t (mod[ 1]) 3, z(,1) w(,1) 96Pt 8 f,, 0So t11, 0(mod ) ) 3) (,1),1),1) 3Ky 0(mod ) ( ( 1, ) 1, ) 1, ) j 8Ky 0 (,,7, ),1),1) 3So f P 8t 0 6) ( ( ), ) ( ), ) 3( t ) 8t 8P 0 3,, 7) z ( 1, w(1, 96 f,, OH 0(mod ). Ptter: II Isted of () write 17 s 17 ( i)( i) For this choice, fter performig clcultios similr to ptter.i,the correspodig o-zero itegrl solutios to (1) re foud to e, 6, z (, ( w (, (, ( ) ( ( Pge

3 O The Homogeeous Quitic Equtio With Five Ukows.3 Ptter: III Rewrite (3) s 17P v u *1 (7) Assume u 17 (8) Write 1 s 1 [ 17 ][ 17 ] (9) Usig (8) d (9) i (7) d employig the method of fctoriztio, defie ( 17P v) ( 17 )( 17 Equtig rtiol d irrtiol prts, we get P 17 8 v 68 3 Sustitutig the vlues of u d v i (),the o-zero distict itegrl solutios of (1) re s follows., 8 3 3,, 1 3 z (, [17 w (, [17 ][68 ][68, ] 3].3.1 Properties 1. [,1),1) 0J ] is sty umer. 3 ( 3,. 1, ) 1, ) 1, ) S 6t 3(mod 97) 3. z(,1) 6936 f 6P 89So (mod ).,,. (,1),1) 36t 3, is divisile y.. ( 1, ) 1, ) 1, ) 3Ky 3J j 7. Ptter :IV Isted of (9), oe my write 1 s [ 17 ][ 17 ] 1 (10) 16 Sustitutig (8) d (10) i (7) d employig the method of fctoriztio, defie [ 17 ] 17P v [ 17 ] Equtig rtiol d irrtiol prts,we get 17 P 17 3 v Sice our iterest is o fidig itegrl solutios, it is possile to choose d so tht P d v re itegers...1 Choice: 1 Let The B P 17A B AB v 17A B 3AB 7 Pge

4 O The Homogeeous Quitic Equtio With Five Ukows u 68A B Sustitutig these vlues i () the correspodig itegrl solutios to (1) re give y, 8A 3B 3AB 1A B 3AB z ( [68A w ( [68A B B ][17A ][17A B B 3AB] 3AB] 17A B AB NOTE; Suppose we choose B such tht A B 0 the u v.cosiderig u, v to e the geertors of Pythgore trigle,.the its re is represeted y y[ z w].. Choice :II Let ( k ) The P [ 17k 6k ] v [ 17k ] u [ 68k 68k 6] Sustitutig these vlues i (),the o-zero distict iteger solutios of (1) is foud to e, [ 8k 68k 0] y [ 1k 68k ] z [68k w [68k P 68k 6][17k 68k 6k][17k [ 17k 6k ]..3 Choice: III Let ( k ) The P [ k k ] v [ k 8k 3] ] ] u [ 16 k k] The the correspodig o-zero distict itegrl solutios of (1) re give y, [ 3k k 9] y [ k k 3] z [16 k w [16 k k][ k k][ k 8k 3] 8k 3] P [ k k ].Remrk: It is worth metioig here tht,the triple (, y, z) d (, y, w) otied from y of the ove ptters stisfy respectively the followig hyperolic proloids. y ( z ) d y ( w ). III. Coclusio Oe my serch for other choices of solutios to (1) log with the correspodig properties. 7 Pge

5 O The Homogeeous Quitic Equtio With Five Ukows Refereces [1]. L.E.Dickso, History of Theory of Numers, Vol.11, Chelse Pulishig compy, New York (19). []. L.J.Mordell, Diophtie equtios, Acdemic Press, Lodo(1969). [3]. Crmichel,R.D.,The theory of umers d Diophtie Alysis,Dover Pulictios, New York (199) []. M.A.Gopl & A.Vijyshkr, A Iterestig Diophtie prolem 3 3 y z Advces i Mthemtics, Scietific Developmets d Egieerig Applictio, Nros Pulishig House, Pp 1-6, 010. []. M.A.Gopl & A.Vijyshkr, Itegrl solutios of terry quitic Diophtie equtio (k ) y z,itertiol Jourl of Mthemticl Scieces 19(1-), ,(j-jue 010) [6]. M.A.Gopl,G.Sumthi & S.Vidhylkshmi, Itegrl solutios of o-homogeeous Terry quitic equtio i terms of pells 3 3 sequece y y) z Jourl of Applied Mthemticl Scieces,vol.6,No.1,9-6,April.013. [7]. M.A.Gopl,G.Sumthi d S.Vidhylkshmi Itegrl solutios of o- homogeeous quitic equtio with three ukows y y y ( k 3) z Itertiol J. of iovtive Reserch i sciece egieerig d tech.vol.,issue.,90-9,april.013. [8]. S.Vidhylkshmi, K.Lkshmi d M.A.Gopl, Oservtios o the homogeeous quitic equtio with four ukows y z ( y)( y ) w,ccepted for Pulictio i Itertiol Jourl of Multidiscipliry Reserch Acdemy (IJMRA). [9]. M.A.Gopl,S.Vidhylkshmi d A.Kvith O the quitic equtio with four ukows y ( k t ) z w ccepted for pulictio i Bessel J. of Mthemtics10.M.A.Gopl,G.Sumthi d S.Vidhylkshmi, Itegrl solutios of the o--homogeeous quitic equtio with four ukows Bessel J. of Mthemtics,3(1),17-180,003. y ( y ) z w z z(1 7w ) [10]. M.A.Gopl & A.Vijyshkr, Itegrl solutios of o-homogeeous quitic equtio with five ukows, Bessel J.Mth.,1(1),3-30,011. [11]. M.A.Gopl & A.Vijyshkr, solutios of quitic equtio with fiveukows 3 y ( z w ) P,Accepted for Pulictio i Itertiol Review of Pure d Applied Mthemtics. [1]. M.A.Gopl, G. Sumthi & S.Vidhylkshmi, O the o-homogeous quitic equtio with five ukows y z w 6T Itertiol Jourl of Mgemet, IT d Egieers.Vol.3,Issue ,April.013 [13]. M.A.Gopl,S.Mllik,d S.Vidhylkshmi, O the o-homogeeous quitic equtio with five ukows 3 Itertiol Jourl of Iovtive Reserch I Sciece,Egieerig d y ( k s )( z w ) p Techology,Vol..Issue ,,April Pge