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1 Ieraioal joral of Egieerig Reearch-Olie Peer Reviewed Ieraioal Joral ricle available olie h:// Vol.1. Ie RESERCH RTICLE ON TERNRY QUDRTIC EQUTION M..GOPLN S.VIDHYLKSHMI S.NIVETHITH Dearme of Mahemaic Shrimai Idira Gadhi College Tirchiraalli Idia ricle Received: 6/10/01 ricle Revied o: 6/1/01 ricle cceed o:/1/01 STRCT The erar qardraic diohaie eqaio give b i aaled for i aer of o-ero diic iegral olio. few iereig relaio bewee he olio ad ecial olgoal mber are ehibied. KEY WORDS: Terar qadraic Iegral olio. M.SC 000 mahemaic bjec claificaio: S.NIVETHITH SCN QR CODE FOR MORE DETILS INTRODUCTION The erar qadraic diohaie eqaio offer a limied field for reearch becae of heir varie [1- ]. For a eeive review of vario roblem oe ma refer [4-1]. Thi commicaio cocer wih e aoher iereig erar qadraic eqaio for deermiig i ifiiel ma oero iegral olio. lo a few iereig relaio amog he olio have bee reeed. NOTTIONS USED m Polgoal mber of rak wih ide m. Peagoal ramidal mber of rak. 16 M..GOPLN e al
2 Ieraioal joral of Egieerig Reearch-Olie Peer Reviewed Ieraioal Joral ricle available olie h:// Vol.1. Ie Terahedral mber of rak.. METHOD OF NLYSIS The erar Qadraic Diohaie Eqaio o be olved for i o-ero diic iegral olio i (1) The biio of liear raformaio v v( v 0) () i (1) lead o v () () i olved hrogh differe aroache ad h differe aer of olio of (1) are reeed below..1 PTTERN-I Wrie a ( i )( i ) (4) me a b () where ab are o-ero diic ieger. Uig (4) ad () i () ad emloig he mehod of facoriaio defie ( i v) ( i )( a i Eqaig real ad imagiar ar we ge v ( a v( a a a b 6b 6ab 4ab Sbiig ad v vale i () we ge ( a a 9b ab ( a a b 10ab Th () (6) ad () reree o-ero diic iegral olio of (1) i wo arameer. few iereig roerie oberved are a follow: a 1) (mod1) ( a ( 1b ) 4 b 1 ( a1) (1 18 0(mod ) a 0(mod ) 1 M..GOPLN e al 8. PTTERN II Irodcig he liear raformaio X T v X T ) i () lead o X W 1T (9) which i aified b T r X 1r W 1r Sbiig he above vale of Tw i ) & () he correodig o-ero diic iegral olio are ( r ( r ( r 6r 1r 1r 14r 14r 6r (6) ()
3 Ieraioal joral of Egieerig Reearch-Olie Peer Reviewed Ieraioal Joral ricle available olie h:// Vol.1. Ie ( r1) 8 i a erfec qare r r ( r 11) 4 r ( r r ) 44 ( 1 ) 4 r Noe: oe iead of ) we have X T v X T 0(mod ) 0(mod10) 0(mod ) Th he correodig o-ero diic iegral olio o (1) are give b ( r ( r ( r 6r 1r 1r 14r 14r 6r ( 18) 16 1 ( r6) ( r 1) 10 ( 4) ( 46. PTTERN III Wrie () i he form of raio a ( v) 0 v which i eqivale o he em of eqaio v v ( ( ) 0 0 0(mod 49) r 84 0(mod108) 0(mod 0) lig he mehod of cro mlilicaio bewee he above em of eqaio we have 6 6 v 4 (1) Sbie (1) i () we have (11) (1) ( 10 (14) ( 9 Th (1) ad (14) reree o-ero diic iegral olio of (1) i wo arameer. ( a ) i a erfec qare. 18 a 164 a ( 0 b 0(mod 4) ( a1) ( 18 a 0(mod ) 18 M..GOPLN e al
4 Ieraioal joral of Egieerig Reearch-Olie Peer Reviewed Ieraioal Joral ricle available olie h:// Vol.1. Ie PTTERN IV () i alo wrie i he form of raio a ( v) ( v) 0 Followig he rocedre a i aer III he correodig Iegral olio of (1) a give b ( ( ( ( 8 b 1) b 9 ( 10 ( a ) ( 1 ( a8) ( 6. PTTERN V Eqaio () i wrie a Wrie 1 a v a ) 0(mod 8) a 1 9 ( 0(mod 4) 0(mod161) )( 4 ) 1 (16) me a b (1) Sbiig (16) & (1) i (1) ad emloig he mehod of facoriaio defie Which i aified b ig he vale of & v i () we have or aim i o fid iegral olio.i i oible o chooe a ad b o ha ad are ieger. Cae (i) Le a b The correodig ieger olio are 19 M..GOPLN e al
5 Ieraioal joral of Egieerig Reearch-Olie Peer Reviewed Ieraioal Joral ricle available olie h:// Vol.1. Ie ( ( ( ( 168 a (mod196) ( (1 110 a 49 0(mod ) ( a8) ( 4 a 11 0(mod1) Cae (ii): Le a 1 b 1. The correodig ieger olio are REMRKLE OSERVTIONS 1. If he o-ero ieger rile ( ) i a olio of (1) he he rile ( ) Where alo aifie (1). Here. If he o-ero ieger rile ( ) i a olio of (1) he he rile ( ) Where alo aifie (1). 0 M..GOPLN e al
6 Ieraioal joral of Egieerig Reearch-Olie Peer Reviewed Ieraioal Joral ricle available olie h:// Vol.1. Ie Here ) ) ) ).... i a erfec qare 4. 0(mod ). 6. If are ake a he geeraor of a Phagorea riagle he he Hoee i cogre o he rodc of i geeraor der modle.. Coider ad o be he legh ad breah of a recagle R whoe rea = Perimeer = P ad Legh of he diagoal= L The i i oed ha 1. L.4( L. P ) 1 0(mod ) 0(mod 8) 4. CONCLUSION I hi aer we have obaied ifiiel ma o-ero diic iegral oi o he homogeeo coe give b To coclde oe ma earch for he iegral oi o oher choice of crve amel herboloid araboloid ad herbolic araboloid ad o o. REFERENCES [1]. L.E.Dicko Hior of Theor of Nmber vol.chelea Pblihig Coma New York 19. []. L.J.Mordell Diohaie Eqaio cademic re Lodo []. dre weil Nmber heor: roach hrogh hior: from hammrai o legedre / dre weil: oo (irkahaer boo)198. [4].M..Goala S.Vidhalakhmi ad S.Devibala Iegral olio of ka ( ) b 4k llei of re ad alied ciece vol.e No: (006) []. M..Goala S.Vidhalakhmi ad S.Devibala Iegral olio of 8 9 Pre ad lied Mahemaika Sciece Vol.LXVI No: [6]. M..Goala ad S.Vidhalakhmi obervaio o ka b c ca Cieica Idica Vol.XXXIIIM No: M..GOPLN e al
7 Ieraioal joral of Egieerig Reearch-Olie Peer Reviewed Ieraioal Joral ricle available olie h:// Vol.1. Ie []. M..Goala Maj omaah ad N.Vaiha Iegral olio of Ciecia Idica Vol.XXXIIIM No:4 (00) k m( ) ca [8].M..Goala ad V.Podichelvi O erar Qadraic Eqaio 1 Imac.J.Sci.Tech Vol() No: [9]. M..Goala ad J.Kaligarai O Terar Qadraic Eqaio 8 Imac J.Sci.Tech; Vol.() No: [10].M..GoalaS.Vidhalakhmi E.Premalaha o erar Qadraic Diohaie Eqaio Dioha J.Mah. 1(1) (01) 1-. [11].M..Goala S.Vidhalakhmi N.Thiriraielvi obervaio o 1 Dioha J.Mah 1() (01) -8. [1].M..GoalaS.vidhalakhmi M.Majla o erar Qadraic Eqaio 19 Dioha J.Mah 1() (01) [1].S.VidhalakhmiK.LakhmiM..GoalaLaice oi o he Elliic araboloid eel J.Mah.() M..GOPLN e al
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