degree non-homogeneous Diophantine equation in six unknowns represented by x y 2z
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1 Scholrs Jourl of Egieerig d Techology (SJET Sch. J. Eg. Tech., ; (A:97- Scholrs Acdeic d Scietific Pulisher (A Itertiol Pulisher for Acdeic d Scietific Resources ISSN -X (Olie ISSN 7-9 (Prit Reserch Article O The Higher Degree Equtio with Six Ukows x 6 -y 6 -z = T (w -p MA Gopl *, S Vidhylkshi, E Prelth, Professor, Deprtet of Mthetics, SIGC, Trichy-6, Tildu, Idi; Assistt Professor, Deprtet of Mthetics, Ntiol college, Trichy-6, Tildu, Idi *Correspodig uthor Dr. M. A. Gopl Eil: Astrct: We presets o zero solutios of the th ( degree o-hoogeeous Diophtie equtio i six 6 6 ukows represeted y x y z T ( w p i which, z. I prticulr, differet ptters of o-zero itegrl solutios of the ove equtio log with few iterestig properties og the solutios re exhiited. Keywords: Higher degree equtio with six ukows, Itegrl solutios INTRODUCTION Diophtie equtios, hoogeeous d o-hoogeeous hve roused the iterest of uerous theticis sice tiquity s c e see fro [,.The prole of fidig ll iteger solutios of Diophtie equtio with three or ore vriles d degree t lest three, i geerl presets good del of difficulties. There is vst geerl theory of hoogeeous qudrtic equtios with three vriles [-7.Cuic equtios i two vriles fll ito the theory of elliptic curves which is very developed theory ut still iportt topic of curret reserch [8-. A lot is kow out equtios i two vriles i higher degrees. For equtios with ore th three vriles d degree t lest three very little is kow. It is worth to ote tht udesirility ppers i equtios, eve perhps t degree four with firly sll co-efficiets. It sees tht uch work hs ot ee doe i solvig higher order Diophtie equtios. I [- few higher order equtios re cosidered for itegrl solutios. I this th couictio ( degree o-hoogeeous equtio with six vriles represeted y 6 6 x y z preseted. T ( w p NOTATIONS USED t, - Polygol uer of rk with size. P - Pyridl uer of rk with size. g - Gooic uer of rk so - Stell octgulr uer of rk pr - Proic uer of rk Pt - Pttope uer of rk is cosidered d i prticulr few iterestig reltios og the solutios re f, s - -diesiol figurte uer of rk with s sides. METHOD OF ANALYSIS The diophtie equtio represetig the Higher degree equtio with six ukows uder cosidertio is 6 6 x y z T ( w p ( 97
2 Gopl MA et l., Sch. J. Eg. Tech., ; (A:97- Itroductio of the trsfortio x u v, y u v, z uv, w uv p uv ( i ( leds to u v T ( Now, we solve ( through differet ethods d thus oti differet ptters of solutios to ( PATTERN -I Assue T T( ( d re o zero distict itegers Write s ( i( i ( Usig ( & ( i ( d pplyig the ethod of fctoriztio, defie u iv ( i ( i ( i( i, sy Equtig the rel d igiry prts, we hve u v Hece i view of (, the correspodig solutios of ( re give y [( i ( i [( i ( i i Illustrtio-I Let =, = Thus the correspodig o-zero distict itegrl solutios of ( re x x( 7 y y( z z( ( w w( ( p p( ( T T ( A few iterestig properties oserved re s follows: z. [ (, w(, p( fors Pythgore triple.. x (, ( 7y(, ( SO ( t, ( ( ( x ( y ( z( w( p( 6. x( 7y( T( t } is Nsty uer. {,. {7 x( y( 7( g } is cuicl iteger x z ( ( w ( ( p ( ( y 98
3 Gopl MA et l., Sch. J. Eg. Tech., ; (A:97- PATTERN-II: Isted of (, write s ( i( i (6 Followig the procedure siilr to Ptter-I d perforig few clcultios, the correspodig o-zero distict x itegrl solutios of ( re foud to e [( i ( i [( i ( i i z ( ( w ( ( p ( ( y Illustrtio-II Let =, = The correspodig o-zero distict itegrl solutios of ( re x x( 6 9 y y( 9 z z( ( ( w w( ( ( 9 p p( ( ( 6 T T( Properties:. w ( p( z( 9 * *. y x( f 8 f 8Pr t 7g (,6,, 9 x ( y( ( f, 8 f,6 6t, (od.. 9x (, y(, SO (od. x ( y ( z( w( p( 6. {y( 9x( t } is iqudrtic iteger. 8, A PATTERN-III: I dditio to ( & (6, write s ( i( i Followig the procedure siilr to Ptter-I, d perforig few clcultios, the correspodig o-zero distict itegrl solutios of ( re give y 99
4 Gopl MA et l., Sch. J. Eg. Tech., ; (A:97- x z ( ( w ( ( y p ( ( [( i ( i. [( i ( i i. Illustrtio-III Let =, = The correspodig o-zero distict itegrl solutios of ( re x x( 6A 6B 9AB y y( A z z( (A w w( (A p p( (A B B B B AB A(A A(A A(A B B B T T ( ( A B Properties:. 9x( t, t, A y( t, t, A Pt A. A A A 9x ( A( A y(( A( A T( A( A P. x ( y ( [ w( [ p(. x ( 9y( t (od., B A (od {9 x( ( A ( A y( ( A ( A T( A } is sty uer: PATTERN-IV: Sustitutig = i (, we hve u v (7 Applyig the ethod of fctoriztio, the correspodig o-zero distict itegrl solutios of (7 re give y u [( i ( i (8 v [( i ( i i Tkig = i (,we hve u v T (9 Cosiderig T T( d eployig the ethod of fctoriztio, the correspodig o-zero distict itegrl solutios of (9 re give y u u v v u v The repetitio of the ove process leds to the solutios of ( represeted y P A
5 Gopl MA et l., Sch. J. Eg. Tech., ; (A:97- u ( iau Bv i v ( Bu iav i A ( i ( i B ( i ( i Hece, the correspodig o-zero distict itegrl solutios of ( re give y x {( iau Bv ( Bu iav } i y {( iau Bv ( Bu iav } i z {( iau Bv ( Bu iav } w {( iau p {( iau T Bv ( Bu Bv ( Bu iav } iav } CONCLUSION I lier trsfortios (,the vriles w d p y lso e represeted y w u v, p u v Applyig the procedure siilr to tht of ptters I to IV, other choices of itegrl solutios to ( re otied. To coclude, oe y serch for other ptters of solutios d their correspodig properties. Mthetics Suject Clssifictio: D99 REFERENCES. Dickso LE; History of Theory of Nuers, Vol., Chelse Pulishig Copy, New York, 9.. Mordell LJ; Diophtie equtios, Acdeic Press, Lodo, x y z. Gopl MA, Vidhlkshi S, Devil S; Itegrl solutios of. Act cici Idic 6; XXXIIM (: Gopl M Sgeeth G; A rerkle oservtios o y x. Ipct.J.sci.Tech, ; (:- 6.. Gopl MA; Note o the Diophtie equtio x xy y z,act cieci Idic.; XXXVIM (: Gopl M Sgeeth G; O the terry cuic Diophtie equtio y Dx z. Archiedes Jourl of Mthetics, ;(: Gopl M Soth M, Vith N; Terry Cuic Diophtie Equtio x y z. Advces i Theoreticl d Applied Mthetics, ; (: Gopl M Soth M, Vith N; Terry cuic Diophtie Equtio (x y z. Act cieci Idic 8;XXXIV M(: Gopl M Sgeeth G; Itegrl solutios of terry o hoogeeious iqudrtic equtio x x y y z z. Accepted i ct cieci Idic ; XXXVII M(:799-8.
6 Gopl MA et l., Sch. J. Eg. Tech., ; (A:97-. Gopl M Soth M, Sgeeth.G, Itegrl solutios of o hoogeeious qurtic equtio x y (k (z w. Archiedes Jourl of Mthetics, ;(:-7.. Gopl MA, Sgeeth G; Itegrl solutios of Terry quitic Diophtie equtio x y (k z. Bulleti of pure d pplied scieces, ; 9E (:-8.. Gopl MA, Jki G; Itegrl solutios of (x y (x y xy (z w, Ipct. J. sci. Tech, ;(:97-.. Gopl MA, Vijyshkr A; Itegrl solutios of terry quitic Diophtie equtio x (k y z. Itertiol Jourl of Mtheticl Scieces, ; 9(-: Gopl M Sgeeth G; O the Sextic Diophtie equtio with three ukows 6 X XY Y (k z, Ipct.j.sci.Tech., ; (: Gopl M Sgeeth G; O the Heptic Diophtie equtio with five ukows x y (X Y z. Atrctic J. Mth., ; 9(: Gopl M Srikth R, Skrry MG, O the Diophtie Equtio Ax Bx y cy Dz, Applied sciece Periodicl, 999; (: Gopl M Sgeeth G; Itegrl solutios of th degree o hoogeeous equtio with three ukows ( x y xy ( ( k z. South est Asi J. Mth.& Mth. Sc, ;9(: Gopl MA, Vijyskr A; Itegrl solutios of ( x y xy ( k z, Ipct. J. sci. Tech, ;(: Mjusoth G, Sgeeth G, Gopl MA; Oservtios o the higher degree Diophtie equtio x y ( k z, Ipct. J.sci. Tech, ; (: Gopl M Vidhlkshi S, Lkshi K; Oservtios o the higher degree Diophtie equtio x y z w,idi jourl of sciece, ; (:-.
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