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1 Ieraioal joural of Egieerig Research-Olie A Peer Reviewed Ieraioal Joural Aricles available olie hp:// Vol.., Issue.., 3 RESEARCH ARTICLE INTEGRAL SOLUTION OF 3 G.AKILA, M.A.GOPALAN, S.VIDHYALAKSHMI 3 M.Phil sude, Deparme of Mahemaics, SIGC, Trich, Tamiladu,Idia,3 Professor, Deparme of Mahemaics, SIGC, Trich, Tamiladu,Idia Aricle Received: // Aricle Revised o:3// Aricle Acceped o:// M.A.GOPALAN ABSTRACT The erar quadraic homogeeous equaio represeig homogeeous coe give b 3 is aaled for is o-ero disic ieger pois o i. Three differe paers of ieger pois saisfig he coe uder cosideraio are obaied. A few ieresig relaios bewee he soluios ad special umber paers amel Polgoal umber, Pramidal umber, Ocahedral umber, Proic umber,sella ocagular umber, Peaope umber ad Nas umber are preseed. Also kowig a ieger soluio saisfig he give coe, hree riples of iegers geeraed from he give soluio are ehibied. Kewords: Terar homogeeous quadraic, iegral soluios Mahemaics Subjec Classificaio: D9 Noaios: m P - Pramidal umber of rak wih sie m. T m, - Polgoal umber of rak wih sie m. Pr - Proic umber of rak G.AKILA INTRODUCTION The erar quadraic Diophaie equaios offer a ulimied field for research due o heir varie [,. For a eesive review of various problems, oe ma refer [-. This commuicaio cocers wih e aoher ieresig erar quadraic equaio 3 represeig a coe for deermiig is ifiiel ma o-ero iegral pois. Also, a few ieresig relaios amog he soluios are preseed. 7 G.AKILA e al

2 Ieraioal joural of Egieerig Research-Olie A Peer Reviewed Ieraioal Joural Aricles available olie hp:// Vol.., Issue.., 3 METHOD OF ANALYSIS: The erar quadraic equaio uder cosideraio is 3 To sar wih i is see ha he riples k,k,k,k,k k,k k ad rs, r 3s, r 3s Saisf. However, we have oher choices of soluios o which are illusraed below: cosider as Wrie as Assume i 3 a 3b 3 3 i Subsiuig 3 ad i ad emploig he mehod of facoriaio. defie i 3 a i [ 3b i Equaig he real ad imagiar pars i he above equaio, we ge [ [ ab a a 3b 3b 3 86ab 3 Replacig a b soluios o are give b A, b b B i he above equaio correspodig ieger [ A {AB [ A {86AB } A A } For simplici ad clear udersadig, akig = i A, he correspodig ieger soluios of are give b A 96B 9 AB 6 A 9866 B 89 AB Properies: A, 6, A A,, A, A 6,, 8, mod 9 mod pr 88 p, mod 879 mod 6 7 G.AKILA e al

3 Ieraioal joural of Egieerig Research-Olie A Peer Reviewed Ieraioal Joural Aricles available olie hp:// Vol.., Issue.., 3 A, 96, A 63mod3, 388, 89 pr Each of he followig represes a as umber 66 A, A 6A A,A 6A, 6mod879 96, p, I is worh o oe ha i ma also be represeed as {3 i 3}{3 i 3} 3 Followig he aalsis preseed above, he correspodig ieger soluios o are foud o be 3 [ A {AB3 3 [ A 3 {3 AB} B 3 A } 6mod38 For he sake of simplici, akig = i B,he correspodig ieger soluio of are give b 88 A 88 B 3666 AB 833 A 7889 B 668 AB 7 A Properies: A, 378, A A, 9, A, 378,, 79, 376mod mod pr 733 p, A, 3668, A 739mod336, 3668,, 397, pr 88mod p, 88mod mod 7339 Geeraio of ieger soluios Le,, be a give ieger soluio of The, each of he followig riples of iegers saisfies : Triple :,, Triple :,, 833mod 98 7 G.AKILA e al

4 Ieraioal joural of Egieerig Research-Olie A Peer Reviewed Ieraioal Joural Aricles available olie hp:// Vol.., Issue.., 3 [{3 } { } [{3 3 } {3 } Triple 3:,, [{3 } { } [{3 3 } {3 } 3 3, Triple :, [{9 } {3 3 } 8 [{3 3 } {9 } 8 CONCLUSION I his paper, we have preseed hree differe paers of o-ero disic ieger soluios of he homogeeous coe give b 3 73 G.AKILA e al.to coclude, oe ma search for oher paers of soluio ad heir correspodig properies. REFERENCES [. Dickso, L.E., Hisor of Theor of Numbers, Vol.,Chelsea Publishig compa, NewYork, 9 [. Gopala, M.A.,Padichevi, V., Iegral soluio of erar quadraic equaio, Acociecia Idica, Vol. XXXIVM, No.3, 33-38,8. [3. Gopala, M.A.,Kaliga Rai, J., Observaio o he Diophaie equaio, J.sci ech ; Vol, 9-9, 8. D Impac [. Gopala, M.A.,Padichevi, V., o erar quadraic equaio, Impac J.sci ech; Vol, -8, 8. [. Gopala, M.A., Maju somaah, Vaiha,N., Iegral soluios of erar quadraic Diophaie equaio k. Impac J.sci ech; Vol, 7-78, 8. [6. Gopala, M.A., Maju somaah, Iegral soluio of erar quadraic Diophaie equaio AaricaJ,Mah,-,8. [7. Gopala,M.A.,ad Gaam,A., Phagorea riagles ad special polgoal umbers, Ieraioal Joural of Mahemaical Sciece, Vol.9,No.-,-, Ja-Ju. [8. Gopala, M.A., ad Vijaasakar, A.,Observaios o a Phagorea problem, Aca Ciecia Idica, Vol.XXXVIM, No.,7-,. [9. Gopala.M.A., ad padichelvi.v., Iegral soluios of erar quadraic equaio, Impac J.sci TSech; Vol,No.,-6,.

5 Ieraioal joural of Egieerig Research-Olie A Peer Reviewed Ieraioal Joural Aricles available olie hp:// Vol.., Issue.., 3 [. Gopala, M.A.,Kaliga Rai, J.O erar quadraic equaio 8,Impac J.sci ech ; Vol, o.,39-3,. [. Gopala, M.A., Geeha, D., Laice pois o he hperbolid of wo shees 6 6, Impac J.sci ech ; Vol,No.,3-3,. [. Gopala, M.A., Vidhalakshmi, S., ad Kaviha, A., Iegral pois o he homogeeous coe 7, DiophausJ.Mah.,,7-36,. [3. Gopala, M.A., Vidhalakshmi, S., Sumahi,G., Laice pois o he hperboloid oe shee 3, DiophausJ.mah.,,9-,. [. Gopala, M.A., Vidhalakshmi, S., ad Lakshmi,K., Iegral pois o he hperboloid of wo shees 3 7, DiophausJ.mah.,,99-7,. [. Gopala, M.A., ad Srividha,G., Observaios o.. [6. Gopala,M.A., Sageeha,G.,Observaio o ArchimedesJ.Mah,, 7-3 AarcicaJ.Mah,9, 39-36,. [7. Gopala,M.A., ad Vijaalakshmi,R., O he erar quadraic equaio, >, Bessel J.Mah,,7-,. [8. Majusomaah, Sageeha,G., Gopala,M.A., O he homogeeous erar quadraic Diophaie equaio k k, Bessel J.Mah,,7-,. [9. Majusomaah, Sageeha,G., Gopala,M.A., Observaios o he erar quadraic equaio 3, Bessel J.Mah,,-,. [. Diva,S., Gopala,M.A., Vijaalakshmi,S.,laice pois o he homogeeous coe, The Eperime,3 vol,73,9-99. [. Mordell, L.J., Diophaie equaios, Academic press, New York, G.AKILA e al