OBSERVATIONS ON TERNARY QUADRATIC DIOPHANTINE EQUATION - x 63y
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1 OBSERVATIONS ON TERNARY QUADRATIC DIOPHANTINE EQUATION - x 63 K.Lakshmi R.Someshwari Asst.Professor, Departmet of Maematics, Shrimati Idira Gadhi College,Tamil adu,idia, M.Phil Scholar, Departmet of Maematics, Shrimati Idira Gadhi College,Tamiladu,Idia, Abstract The terar quadratic homogeeous equatio represetig homogeeous coe give b x 63 is aalsed for its o-ero distict iteger poits o it. Three differet patters of iteger poits satisfig e coe uder cosideratio are obtaied. A few iterestig relatios betwee e solutios ad special umber patters, amel, Polgoal umber, Pramidal umber, Octahedral umber, Proic umber, Decagoal ad Nast umber are preseted. Also, kowig a iteger solutio satisfig e give coe, four triples of itegers geerated from e give solutio are exhibited. Kewords Terar quadratic Diophatie equatios, polgoal umber, pramidal umber, octahedral umber, proic umber, decagoal umber. Msc Subject Classificatio: D9. I. INTRODUCTION The terar quadratic Diophatie equatios offer a ulimited field for research due to eir variet [-]. For a extesive review of various problems, oe ma refer [-]. This commuicatio cocers wi et aoer iterestig terar quadratic equatio x 63 represetig a coe for determiig its ifiitel ma o-ero iteger poits. Also, a few iterestig relatios amog e solutios are preseted. NOTATIONS: P m - Pramidal umber of rak wi sie m. T m, - Polgoal umber of rak wi sie m. Pr - Proic umber of rak. OH - Octahedral umber of rak. T, - Decagoal umber of rak. II. METHOD OF ANALYSIS The terar quadratic equatio uder cosideratio is x 63 ( To start wi it is see at e triples ( k,3k,3k, (k,k k 3, k k 3 ad (rs, r 63s, r 63s satisf (. However, we have oer choices of solutios to ( which are illustrated below. Cosider ( as x 63 * ( Assume ( a, b a 63b All Rights Reserved 99
2 Patter: Write as Volume, Issue 6; Jue - 6 [ISSN: -7] (3 i( 63 (3 i( 63 ( (3 Substitutig (3 ad ( i ( ad emploig e meod of factoriatio, defie (3 i( 63 ( x i 63 ( a i 63b (3 Equatig e real ad imagiar parts i e above equatio, we get ( a 63b (3 ( 6ab x ( (3 (3 ab( a 63b ( (6 (3 Replacig a b ( 33 A ad b b ( 33 B i e above equatio e correspodig iteger solutios to ( are give b x ( 3(3 ( A 63B 6( (3 ( ( 3( A 63B (3 (3 ( 3 ( A 63B For simplicit ad clear uderstadig takig = i (A e correspodig iteger solutios of ( are give b x 99 A 696 B 3 AB 3A 6B 98AB A 6 B Properties: x( A, 676(mod3 t986, A x (, t3, 3 pr 99(mod99 3 x (, t 696t 86p 99(mod ,, ( 66, A (, t397, 98 pr (, 6t 3968p, t66, ( A, t, A 63(mod3 A, t (mod 3(mod99 6 3(mod 9 7 Patter: It is wor to ote at i ( ma also be represeted as (63 i 63 (63 i 63 (63 Followig e aalsis preseted above, e correspodig iteger solutios to ( are foud to be x (63 (63 ( A 63B (63 AB (63 ( A 63B (63 (63 AB (B (63 ( A 63B AB AB All Rights Reserved
3 Volume, Issue 6; Jue - 6 [ISSN: -7] For e rate of simplicit takig = i (B, e correspodig iteger solutios of ( are give b x 393A 939 B AB 68A 688B 796AB 89A 887B Properties: x( A, (mod986 t798, A x (, t97, pr 393 (mod x (, t 939t 6736p 393(mod ,, ( 38, A (, t333, 796pr (, 688t 8p, t38, ( A, t898, A 63(mod88 A, t 38(mod873 68(mod (mod 83 7 III. REMARKABLE OBSERVATIONS If e o-ero iteger triple ( x,, is a solutios of ( e each of e followig ree triples also satisf (. Triple : ( x,, Let e first solutio of ( be x x 6h h h Substitutig i (, we get ( h ( x 6h 63( h h h36h hx 63h 6 h h x 6 The we get e first solutio, x 73x 76 x 7 x 6 99 Repeatig e above process, oe ma get ma iteger solutios. Triple : ( x,, Let e first solutio of ( be x 3x h 3 3 h Substitutig i (, we get ( 3 h (3x h 63(3 h h 6 h hx (7 All Rights Reserved
4 Volume, Issue 6; Jue - 6 [ISSN: -7] 3h 6x h x Substitutig h value i (8, we get x 3x x x x 3 (x x Hece e matrix represetatio of e above solutio is x x From e first solutio we get e correspodig secod solutio as give below: x x (x (x x 9 x x (x (x 9 The matrix represetatio is x 9 x 9 Similarl to fid e solutio of (, let A To fid Eige values of A: Cosider, A I ( ( Eige values are 3 ad -3 Take 3, 3 To fid A,we use e followig formula A AI AI (9 Substitutig, ad A values i (9,we get 3 8 ( 3 A 6 ( (8 ( 3 ( 3 ( ( 3 ( A 6 3 ( ( 3 ( 3 ( ( 3 All Rights Reserved
5 x 6 Volume, Issue 6; Jue - 6 [ISSN: -7] 3 (8 ( 3 ( 3 ( ( 3 ( 3 ( ( 3 ( 3 ( ( 3 (8 The we get solutio x 3 (8 ( 3 ( x 6 3 ( ( 3 ( ( ( 3 ( x 3 ( ( 3 (8 x Triple 3: ( x,, Let e first solutios of ( be x 3x 3 h h 3 Substitutig i (, we get ( 3 h (3x 63(3 h h 6h 63h 396h 6h h 63 Substitutig h value i (,we get Hece e matrix represetatio of e above solutio is From e first solutio we get e correspodig secod solutio as give below: ( 3 ( ( ( 3 96 The matrix represetatio is Similarl to fid e solutio of (, let 3 A 63 3 To fid Eige values of A: All Rights Reserved 3
6 Cosider, A I ( 3 ( Eige values are 3 ad -3 Take, ad A values i (9,we get 3 ( 3 63 A ( 6 63 (3 ( ( 3 (63 A 6 (3 ( 63 ( 3 (63 3, Substitutig 3 Volume, Issue 6; Jue - 6 [ISSN: -7] (3 ( ( 3 ( (3 (63 ( 3 ( (3 ( ( 3 (63 (3 ( ( 3 ( 6 (3 ( 63 ( 3 (63 (3 (63 ( 3 ( The we get x ( 3 x (3 6 (3 6 solutio ( ( 3 (63 (3 ( ( 3 ( ( 63 ( 3 (63 (3 (63 ( 3 ( Triple : ( x,, Let e first solutios of ( be x 3x h 3 h Substitutig i (,we get ( (3x h 63(3 h h 6hx 3 h 6 6h 6x 3 h x 63 Substitutig h value i (, we get x 3x ( x 63 x x 63 3 x x Hece e matrix represetatio of e above solutio is x 3 63x 3 All Rights Reserved
7 Volume, Issue 6; Jue - 6 [ISSN: -7] From e first solutio we get e correspodig secod solutio as give below: x 3x 63 3(3x 63 63( 3 x x x 3 x 3( 3 x (3x 63 The matrix represetatio is x x Similarl to fid e solutio of (, let 3 63 A 3 To fid Eige values of A: Cosider, A I ( 3 ( Eige values are 3 ad -3 Take 3, 3 Substitutig, ad A values i (9,we get ( ( 3 63 A 6 ( 6 63 (3 (63 ( 3 ( (3 ( 63 ( 3 (63 A 6 (3 ( ( 3 ( (3 ( ( 3 (63 x (3 (63 ( 3 ( (3 ( 63 ( 3 (63 6 (3 ( ( 3 ( (3 ( ( 3 (63 The we get x (3 6 (3 6 solutio (63 ( 3 ( x (3 ( 63 ( 3 (63 ( ( 3 ( x (3 ( ( 3 (63 IV. CONCLUSION I is paper, we have preseted two differet patters of ifiitel ma o-ero distict iteger solutios of e homogeeous coe give b x 63. To coclude, oe ma search for oer patters of solutio ad eir correspodig properties. All Rights Reserved
8 Volume, Issue 6; Jue - 6 [ISSN: -7] REFERENCES. Dickso, L.E., Histor of eor umbers, Vol., Chelsea Publishig compa, Nework, 9.. Gopala, M.A., padiselvi, V., Itegral solutio of terar quadratic equatio (x + = x, Act cieciaidica, Vol. XXXIVM, No.3,33-38,8. 3. Gopala, M.A., Kaligarai, J., Observatio o e Diophatie equatio, = Dx + Impact J.sci tech; Vol (, 9-9, 8.. Gopala, M.A., Padiselvi, V., O e terar quadratic equatio x + = +, Impact J.sci tech; Vol (, - 8,8.. Gopala, M.A., Majusomaa, Vaia, N., Itegral solutios of terar quadratic Diophatie equatio x + = (k +. Impact J.sci tech; Vol (, 7-78, Gopala, M.A.,Majusomaa, Itegral solutio of terar quadratic Diophatie Equatio x = = x. Atartical, ma, -, ( Gopala, M.A., ad Gaam,A., Pagorea triagles ad special polgoal umbers, Iteratioal joural of Maematical Sciece, Vol.(9,No.-,-, Ja-Ju. 8. Gopala, M.A., ad Vijaasakar, A. Observatios o a Pagorea problem, ActacieciaIdica, Vol.XXXVIM, No.,7-,. 9. Gopala, M.A., Geea, D., Lattice poits o e hperbolid of two sheets x 6x + + 6x + = +, Impact J.sci tech; vol(,no.,3-3,.. Gopala. M.A., Kaligarai, J. O terar quadratic equatio x + = + 8, Impact J.sci tech;vol (,No.,39-3,.. Gopala, M.A., ad Padiselvi.V., Itegral solutios of terar quadratic equatio (x = x, Impact J.sci tech; Vol (,No.,-6,.. Gopala, M.A.,Vidhalakshmi, S., ad Kavia, A., Itegral poits o e homogeeous coe = x 7, DiophatusJ.Ma.,(,7-36,. 3. Gopala,M.A., Vidhalakshmi, S., Sumai.G., Lattice poits o e hperbolid oe sheet = x + 3, Diophatusj. Ma.,(,9,.. Gopala, M.A.,, Vidhalakshmi S., ad Lakshmi, K., Itegral poits o e hperboloid of two sheets 3 = 7x +, DiophatusJ.Ma.,(,99-7,.. Gopala, M.A., ad Sri vidha, G., Observatios o = x +, ArchimedesJ.Ma,(, 7-,. 6. Gopala, M.A., Sageea, G., Observatio o = 3x, AtarcticaJ.Ma,9(,39-36,,. 7. Gopala, M.A., ad Vidhalakshmi, R., O e terar quadratic equatio x = (α (, α >, Bessel J.Ma,(,7-,. 8. Maju somaa, Sageea, G., Gopala, M.A., O e homogeeous terar quadratic Diophatie equatio x + (k + = (k +, Bessel J.Ma,(,-,. 9. Majusomaa, Sageea, G., Gopala, M.A., Observatios o e terar quadratic equatio = 3x +, Bessel J.Ma,(,-,.. Diva, S., Gopala, M.A., Vidhalakshmi, S., Lattice poits o e homogeeous coe x + =, The Experimet, Vol,7(3, 9-99,3.. Mordell, L.J., Diophatie equatios, Academic press, New All Rights Reserved 6
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