ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01 OSERVTIONS ON THE NON- HOMOGENEOUS SEXTIC EQUTION WITH FOUR UNKNOWNS y ( k z S.Vidhylkhmi 1,M..Gopl,.Kvith Profeor, PG d Reerch Deprtmet of Mthemtic,Shrimti Idir Gdhi college, Trichy-0 00, Tmildu, Idi 1 Profeor, PG d Reerch Deprtmet of Mthemtic,Shrimti Idir Gdhi college, Trichy-0 00, Tmildu, Idi Lecturer, PG d Reerch Deprtmet of Mthemtic,Shrimti Idir Gdhi college, Trichy-0 00, Tmildu, Idi trct: The etic o-homogeeou equtio with four ukow repreeted y the Diophtie equtio y ( k z w i lyzed for it ptter of o-zero ditict itegrl olutio re illutrted. Vriou iteretig reltio etwee the olutio d pecil umer, mely polygol umer, Pyrmidl umer, Jcothl umer, Jcothl-Luc umer, Proic umer, Str umer re ehiited. Keyword: Itegrl olutio, etic o-homogeeou equtio. M.c 000 mthemtic uject clifictio: 11D1 NOTTIONS t, : Polygol umer of rk with ize m m CP, : Cetred polygol umer of rk m Ky m CP,, : Kye umer of rk : Cetred pyrmidl umer of rk F : Four dimeiol figurtive umer of rk whoe geertig polygo i hego F : Four dimeiol figurtive umer of rk whoe geertig polygo i heptgo S,,7 Pr j J : Str umer of rk : Proic umer of rk : Jcothl luc umer of rk : Jcothl umer of rk w I. INTRODUCTION The theory of Diophtie equtio offer rich vriety of fcitig prolem [1-. Prticulrly, i [-, etic equtio with ukow re tudied for their itegrl olutio. [7- lyze etic equtio with ukow for their o-zero iteger olutio. Thi commuictio lye other etic equtio with ukow give y y ( k z w Copyright to IJIRSET www.ijiret.com 101
ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01. Vriou iteretig propertie mog thepropertie mog the vlue of,y,z,w re preeted. II. METHOD OF NLYSIS The equtio uder coidertio i y (1 where k i give o-zero iteger.ce:1 Itroductio of the trformtio ( k z u v, y u v, w u ( I (1 led to u v ( k ( Let z ( z Uig ( i ( d pplyig the method of fctoriztio, defie u i v ( k i ( i ( Equtig rel d imgiry prt, we get u ( ( v ( k ( k ( 0 0 ( Uig ( d ( i (, we hve 1 w i ( ( k ( k ( 0 0 1 ( k 1( 0 ( k ( 0 ( 0 k ( 0 (7 {( k ( 0 ( k 1( 0 } Thu (7 d ( repreet the o-trivil itegrl olutio of (1..Propertie: ( 1 1 (,1 ( k 1[ j 1 0J1 j 7 1 1 1 ( k [( j1 ( 1 0(J 1 ( 1 ( j1 ( 1 ( w (1, {( k [(F,,7 F,, CP 18Pr t, 8 10 7 ( k 1[ t, (CP CP CP 1Pr 1t, Copyright to IJIRSET www.ijiret.com 10 1
ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01 ( y (, 0(mod For illutrtio d cler udertdig, utitutig =1 i (7, the correpodig o-zero ditict itegrl olutio to (1 re give y (k ( ( z( 0 0 [( k ( 0 k( (k ( 0 (k ( 0 0 C.Propertie: 1 ( 1 (1, 1, (k [1 0( ky j1 ( 1 1 (J, ( k [ ( 0( j ( 1 ( ky J ( (1, 1, (k [ 0( ky j1 1J 1 ( k [ ( 0 j ( ky J 8 (,1 ( k [( t, ( CP CP CP 8t, t, 11 (k [F,,7 CP CP 8t, t, 1 ( (1, 1, 1, [( k [t, ( t0, 1CP t1, D.Ce: 1 ( k [ t, (CP CP CP t, 1t, Coider the trformtio u v, y u v, w u (8 Uig (8 i (1, we hve u v ( k z ( Employig the fctoriztio method, defie u i v ( k i ( i Equtig rel d imgiry prt, oe h u k( v ( 0 0 ( k( 0 0 1 1 Thu, the o-zero ditict itegrl olutio to (1 re give y ( k 1( ( k 1( z( k( E.Propertie: 0 0 0 ( ( k ( ( k ( 0 0 0 Copyright to IJIRSET www.ijiret.com 10
ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01 8 ( 1,1 ( k 1[ t, (CP CP CP t, t, 8 ( k [ t, (t1, CP t, 1t, 8 (,1,1 [(t, ( CP CP CP t, t, k[(10t, ( t, CP 0t, ( y (1, 1, [ t, (t, 8t, t, 0t, 1 11 1 k[ t, (CP CP CP 10t, t, 1 8 (,1,1,1 k[ t, ( CP CP CP t, t, 8 [( t, ( t1, CP t, 1t, 8 17 ( 1, ( k 1[ t, ( S CP CP t, t1, 1t, t, F.Ce: Write ( u v (10 Write 1 ( k [(t, ( CP 7CP 10t, t, ( k z *1 (1 i (1 i 1 (11 Sutitutig ( d (11 i (10 d employig the fctoriztio method, defie u i (1 i v ( k i ( i Equtig rel d imgiry prt, we get 1 u [( k ( v [( k 1( 0 ( k 1( ( k ( 0 1 0 0 1 Thu, tkig = = the o zero ditict itegrl olutio to (1 re give y [( k 1( [( z( ( 0 0 [( k ( 0 ( K ( Copyright to IJIRSET www.ijiret.com 10 K( 0 ( K 1( 0 0
ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01 It i to e oted tht oe my get itegrl olutio to (1 whe k tke odd vlue. G.Propertie: (,,,, 1 (1, {( k 1[(F CP t 0t 1 1 ( k [ t, (CP CP t, t, 10t, t, (,1 {[ t, (CP CP CP t, t, 8 k[ 0F,, 10CP 10t, t, (, 8,, } w (1, {( k (1t ( t 0CP 0t 1 7 ( k 1[t, ( CP CP t, 1t, ( 1, 1, 1, {(k [( F,,7 Cp CP 8t, 7t, 1 } } 1 ( k [( CP ( t0, ( t, ( t18, CP t, 1t, } H.Ce: Ited of (11 we write 1 (1 i (1 i 1 Followig the procedure preeted i ptter the correpodig o- zero ditict itegrl olutio to (1 re otied 7 [(k 11( 0 (11k 1( 0 7 [(k 1( z( 7 ( 7 [( k 1( I.Propertie: 0 0 (1k ( (k 1( 0 0 8 8 (,,, 1 (,1 7 [(k 11( t (CP CP CP t t 8 (, 11k 1[(1F,, CP CP, 18t, t 1 10 (,,,, y (1, 7 {(k 1[1t ( CP 1(CP t t 1 ( 1k [t, ( CP CP t, 1t, ( z (, i perfect qure. ( 1, 7 [( k 1{[F,,7 CP CP 1t, 1t, 1 } [ S 1t, t, } (1 k[ CP ( t8, t, t, 10 10t, t, Copyright to IJIRSET www.ijiret.com 10
J.Ce: ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01 Coider the trformtio u v, y u v, w ( u (1 Repetig the proce i ptter 1, the o-zero ditict itegrl olutio to (1 re otied ( ( k k 1( 0 ( k k( 0 y ( ( k k 1( 0 ( k k( z( 0 w ( ( {( k ( 0 (1 k( Where =,, K.Propertie: 0 ( 1 1, ( k k 1[(F,,8 F,, CP t17, 18t, 1t, ( k k[( t, [CP 18t, t, 10t, t, 10 (, y (1, ( k 1 k[1t, ( CP, 1(CP t, t 1 ( k k [( CP ( t8, t, t, 10 10t, t, (,1 ( {( k [ t, (CP CP CP t, t, (1 k[0f,, 10CP 10t, t, 8 } 1 III. CONCLUSION To coclude oe my erch for other ptter of olutio d their correpodig propertie. REFERENCES [1. L.E.Dicko, Hitory of Theory of Numer, Vol.11, Chele Pulihig compy, New York (1. [. L.J.Mordell, Diophtie eequtio, cdemic Pre, Lodo (1. [. Telg,S.G., Numer theory, Tt Mc Grw Hill pulihig compy, New Delhi (1 [. Crmichel, R.D., The theory of umer d Diophtie lyi, Dover Pulictio, New York (1. [. M..Gopl d Sgeeth.G., O the etic equtio with three ukow y y ( k z, Impct J.Sci.tech. Vol., No:, 8-(010. [. M..Gopl.Mju Somth d N.Vith Prmetric Solutio of ct cieci idic XXXIII,,108-108(007. y z, Copyright to IJIRSET www.ijiret.com 10
ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01 [7. M..Gopl d.vijyskr, Itegrl Solutio of the etic equtio y z w, Idi Jourl of Mthemtic d mthemticl ciece, Vol., No, 1-,(010. [8. M..Gopl S.Vidhylkhmi d.vijykr, Itegrl olutio of ho- homogeeou etic equtio y z w, impct j.sci.tech., Vol, No:1, 7-, 01. [. M.. Gopl S.Vidhylkhmi d K.Lkhmi, O the o-homogeeou etic equtio, IJM(, 171-17, Dec.01. ( w y y z Copyright to IJIRSET www.ijiret.com 107