- Pyramidal number of rank n with size m. - Polygonal number of rank n with size m.

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1 Scholars Joural of Egieerig ad Techology SJET Sch. J. Eg. Tech., 0; :9-3 Scholars cademic ad Scietific Publisher Iteratioal Publisher for cademic ad Scietific Resources ISSN 3-35X Olie ISSN Prit Research rticle Lattice Poits o the Homogeeous Coe 59X Y Z K.Meea, S.Vidhyalakshmi, N.Sujitha 3, M..Gopala * Former VC, Bharathidasa Uiversity, Tiruchirappalli, Idia, Professor, Departmet of Mathematics, Shrimati Idira Gadhi College, Tiruchirappalli, Idia 3 Research Scholar i Mathematics, Shrimati Idira Gadhi College, Tiruchirappalli, Idia *Correspodig author Dr. M..Gopala bstract: The terary quadratic homogeeous equatio represetig homogeeous coe give by 59x y z by is aalyzed for its o-zero distict iteger poits o it. Three differet patters of iteger poits satisfyig the coe uder cosideratio are obtaied. few iterestig relatios betwee the solutios ad special umber patters amely Polygoal umber, Pyramidal umber, Octahedral umber, Proic umber, Stella Octagular umber, Petatope umber ad Nasty umber are preseted. lso kowig a iteger solutio satisfyig the give coe, three triples of itegers geerated from the give solutio are exhibited. Keywords: Terary homogeeous quadratic, itegral solutios INTRODUCTION The terary quadratic Diophatie equatios offer a ulimited field for research due to their variety [,. For a extesive review of various problems, oe may refer [ - 0. This commuicatio cocers with yet aother iterestig terary quadratic equatio 59x y z represetig a coe for determiig its ifiitely may ozero itegral poits. lso, a few iterestig relatios amog the solutios are preseted. Notatios m P - Pyramidal umber of rak with size m. T, - Polygoal umber of rak with size m. m Pr - Proic umber of rak OH - Octahedral umber of rak SO - Stella octagular umber of rak Pt - Petatope umber of rak METHOD OF NLYSIS The terary quadratic equatio uder cosideratio is 59x y z To start with it is see that the triples k,9k,30k,k,k k 9,k k 30 ad rs, r 59s, r 59s satisfy. However, we have other choices of solutios to which are illustrated below Cosider as 59x y z ssume Write as z a 59b 3 9

2 Gopala M et al., Sch. J. Eg. Tech., 0; :9-3 {9 i 59}{9 i 59} 30 Substitutig 3 ad i ad employig the method of factorizatio, defie [9 i 59 y i 59x a i 59b 30 Equatig the real ad imagiary parts i the above equatio, we get x 9 ab a 59b 30 y 9 a 59b 8ab 30 z a 59b Replacig a by 30, b by 30 B i the above equatio correspodig iteger solutios to are give by x 30 [ 59B {B9 } y 30 [ 59B 9 {8B } z 30 59B For simplicity ad clear uderstadig, takig i, the correspodig iteger solutios of are give by x B 70B y B 350B z 30 59B Properties x, t mod769 6,, t 3, t8,, t 3,, t 8, t36,, t 7,, t 86, t88,, B t 35, mod 9, B t 066, B, B t 53, B t53, B x, t, y, t 60, mod 90 x, b t 0, b y, b t 3, b x mod768 x mod768 x mod768 y 58mod 67 y 56mod 67 x y 870mod 5869 y 58mod mod mod88 9x, y, 0mod x, B y, B 0mod 5000 z, B t 59, B, t, z mod 6 z t t mod 6,, 8, 0

3 Gopala M et al., Sch. J. Eg. Tech., 0; :9-3 z t t mod 5,,, z, 59t t mod,, 30 y, 68t 6p t56,, 30 y, 68t 6p t37,, 7mod 76 7mod x, y, t 0mod , 30 x, t 36t 7 OH 59mod 3 7,, 30 x, t 36t 58 SO 59mod 36 76,, 30 x, P r t 59mod75 8, 5 30x, t t 6P 59mod76, 0, 30 x, 3 39Pt P 59t 770P r, t598, 0z, is a Nasty umber It is worth to ote that i may also be represeted as {59 i 59}{59 i 59} 59 Followig the aalysis preseted above, the correspodig iteger solutios to are foud to be x 59 [ 59B {B59 } y 59 [ 59B 59 {7B} B z 59 59B For the sake of simplicity, takig i B, the correspodig iteger solutio of are give by x 5 868B 6930B y B 9736B z 63 59B Properties x, t 506mod , 55x, y, 0mod350 55x, B y, B 0mod350 55x, y, t 35, p 55 x, y, t 8690, 0mod x, y, p t686070, 0mod39 55 x, y, p t, 0 0mod x, 3 y, pt 0 63 x, t 6, 36mod x, 55t 6p t600,, 35mod x, 5y, 0mod x, 5y, t 5878, 0mod5875 mod 53 Geeratio of iteger solutios

4 Gopala M et al., Sch. J. Eg. Tech., 0; :9-3 Let x 0, y0, z0 be ay give iteger solutio of The, each of the followig triples of itegers satisfies Triple : x, y, z x x0 y0 6 y x0 9y0 z x0 3y0 x, y, z 9 z 36 3z0 9 7z Triple : 0 x [{ } x0 {9 9 } z0 58 y 9 y 0 z [{ } x0 {599 9 } 58 Triple 3: x, y, z 0 z x3 [{ } x0 { } y0 60 y [{ } x0 { } 60 z3 z 0 Triple : x, y, z 3 y0 x 3 x 0 y [{83 3 } y0 { 3 3 } z0 6 z [{3 3 } y0 {8 3 3 } z0 6 CONCLUSION I this paper, we have preseted three differet patters of o-zero distict iteger solutios of the homogeeous coe give by 59x y z. To coclude, oe may search for other patters of solutio ad their correspodig properties. REFERENCES. Dickso LE; History of Theory of Numbers, Volume, Chelsea Publishig Compay, New York, 95.. Gopala M, Padichevi V; Itegral solutio of terary quadratic equatio z x y xy, cta Ciecia Idica, 008; XXXIVM 3: , 3. Gopala M, Kaliga Rai J; Observatio o the Diophatie equatio, y Dx z, Impact J Sci Tech., 008; : Gopala M, Padichevi V; O terary quadratic equatio a 8b ab, Impact J Sci Tech., 008; : Gopala M, Somaath M, Vaitha N; Itegral solutios of terary quadratic diophatie equatio x y k z. Impact J Sci Tech., 008; : Gopala M, Somaath M; Itegral solutio of terary quadratic Diophatie equatio xy yz zx. tarctica J Math., 008; 5: Gopala M, Gaam ; Pythagorea triagles ad special polygoal umbers. Iteratioal Joural of Mathematical Sciece, 00; 9-: -5.

5 Gopala M et al., Sch. J. Eg. Tech., 0; : Gopala M, Vijayasakar ; Observatios o a Pythagorea problem. cta Ciecia Idica, 00; XXXVIM : Gopala M, Padichelvi V; Itegral solutios of terary quadratic equatio z x y xy. Impact J Sci Tech.; 0; 5: Gopala, M.., Kaliga Rai, J., O terary quadratic equatio x y z 8, Impact J. Sci. Tech.; Vol. 5, No., 39-3, 0.. Gopala M, Geetha D; Lattice poits o the hyperboloid of two sheets x 6xy y 6x y 5 z. Impact J Sci Tech., 00; : Gopala M, Vidhyalakshmi S, Kavitha ; Itegral poits o the homogeeous Coe z x 7y. Diophatus J Math., 0; : Gopala M, Vidhyalakshmi S, Sumathi G; Lattice poits o the hyperboloid oe sheet z x 3y. Diophatus J Math., 0; : Gopala M, Vidhyalakshmi S, Lakshmi K; Itegral poits o the hyperboloid of two sheets 3y 7x z. Diophatus J Math., 0; : Gopala M, Srividhya G; Observatios o 6. Gopala M, Sageetha G; Observatio o y y x z rchimedes J Math, 0; : x z. tarctica J Math, 0; 9: Gopala M, Vijayalakshmi R; O the terary quadratic equatio x y z,. Bessel J Math, 0; : Somaath M, Sageetha G, Gopala M; O the homogeeous terary quadratic Diophatie equatio x k y k z. Bessel J Math, 0; : Somaath M, Sageetha G, Gopala M; Observatios o the terary quadratic Equatio y 3x z. Bessel J Math, 0; : Divya S, Gopala M, Vijayalakshmi S; Lattice poits o the homogeeous coe x y 0z. The Experimet, 03; 73: Mordell LJ; Diophatie equatios. cademic press, New York,