Volume:, Issue: 6, -7 Jue 015.allsubjectjoural.com e-issn: 49-418 p-issn: 49-5979 Impact Factor:.76 M.A. Gopala Professor, Departmet of Idira Gadhi College, Trichy-6000, Tamiladu, Idia. A. Kavitha Lecturer, Departmet of Idira Gadhi College, Trichy-6000, Tamiladu, Idia. G. Thamaraiselvi M. Phil., Scholar, Departmet of Mathematics, Shrimati Idira Gadhi College, Trichy-6000, Tamiladu, Idia. Observatio o the Biquadratic Equatio ith Five Ukos (x-y(x y (k k ( X Y M.A. Gopala, A. Kavith G. Thamaraiselvi Abstract (x, y,x, Y, We obtai ifiitely may o-zero iteger quituples satisfyig the Bi-Quadratic (x - y (x y (k k ( X Y equatio. Various iterestig relatios betee the solutios ad special umbers, amely, Polygoal umbers, Proic umbers, Pyramidal umbers, Stella Octagular umbers, ad Octahedral umbers are exhibited. Keyords: Biquadratic equatio ith five ukos, Itegral solutios, Polygoal ad Pyramidal umbers. 010 Mathematics subject classificatio: 11D5. Notatios Used t m, : Polygoal umber of rak ith sides m t m, m P P m ( 1(m 1 : Pyramidal umber of rak ith sides m ( 1 6 ( m (5 m SO : Stella Octagular umber of rak SO ( 1 Pr : Proic umber of rak Pr OH ( 1 : Octahedral umber of rak OH 1 ( 1 Correspodece: M.A. Gopala Professor, Departmet of Idira Gadhi College, Trichy-6000, Tamiladu, Idia. 1. Itroductio The theory of Diophatie equatios offers a rich variety of fasciatig problems. I particular biquadratic Diophatie equatios, homogeeous ad o-homogeeous have aroused the iterest of umerous mathematicias sice atiquity [1-4] [5-10, 1,. I this cotext oe may refer 1] for various problems o the biquadratic Diophatie equatios ith four ad five variables [11]. have discussed the problem o biquadratic Diophatie equatio ith five ukos. This paper cocers ith yet aother problem of determiig o-trivial itegral solutios of the o-homogeeous biquadratic equatio ith five ukos give by (x - y (x y (k k ( X Y. A fe relatios amog the solutios are preseted. ~ ~
. Method of Aalysis The Diophatie equatio represetig the biquadratic equatio ith five ukos uder cosideratio is (x - y (x y (k k ( X Y Itroducig the liear trasformatios x u v, X u v, y u v Y u v i (1, it simplifies to u v (4k 4k 4 The above equatio ( is solved through differet choices ad thus, oe obtais distict patters of iteger solutios to (1.1 Patter: 1 Let a b (4 Substitutig (4 i ( ad usig the method of factorizatio, defie ( u i v ( a i ((k 1 i Equatig real ad imagiary parts, e have u a b ka 6kb 6ab v a b ab 4kab Substitutig the values of u ad v i (, the o-zero distict itegral solutios of (1 are give by a 6b ka 6kb 4ab 4kab ka 6kb 8ab 4kab a 9b 4ka 1kb 10ab 4kab (5 Y ( a b 4ka 1kb 14ab 4kab ( a b. Properties a 1- a 1 4t 4,a 4Pra ( 1 1a 6 0 a -1 a -1 t46, a 4t6, a (1 -a 6 0 (1 ( ( Write 1 as 1 1 i 1 i 4 Usig (7 i (6 ad employig the method of factorizatio, defie ( u i v ((k 1 i ( a i Equatig real ad imagiary parts i (8, e get 1 i * u a b 6ab ka kb 6kab v a b ab ka kb kab Usig (9 ad (, e get the itegral solutio of (1 as 8ab ka 6kb 4kab a 6b 4ab 8kab a b 14ab ka 9kb 10kab (10 Y ( a 9b 10ab ka kb 14kab ( a b.4 Properties a 1 a 1 16Pr a( k(7t4, a 0a15 0 a 1 ( a 1 4t 4,a 1OH a (1 0.5 Note It is to be oted that i additio to (7, 1 may also represeted as the product of complex cojugates as give belo: Choice: 1 1 i 1 i 1 4 Choice: 11 i4 11 i4 1 169 (7 (8 (9 Choice: 11 i4 11 i4 1 169. Patter: ( ca be ritte as u v (4k 4k 4 *1 (6 ~ 4 ~ Choice: 4 1 i4 1 i4 1 49
Choice: 5 1 i4 1 i4 1 49 Choice: 6 11 i5 11 i5 1 Choice: 7 11 i5 11 i5 1 Choice: 8 1 i 1 i 1 Choice: 9 1 i 1 i 1 Employig the procedure preseted i patter:, the correspodig itegral solutios to (1 for the above choices of 1 are obtaied as belo; Solutio for choice: 1 a 6b 4ab 8kab a 6b 4ab ka 6kb 4kab X ( 4a 1b 4ab ka kb 14kab Y ( 4a 1b 4ab ka 9kb 10kab ( a b 17a 51b 8ab 4ka 6kb 9kab 7a 81b 8ab1ka 1kb 100kab ( 7a 1b Solutio for choice: 5 10a 0b 44ab 6ka 18kb 5kab 16a 48b 8ab 10ka 0kb 44kab a 69b 6ab 4ka 1kb 100kab Y ( 9a 87b 10ab 1ka 6kb 9kab ( 7a 1b Solutio for choice: 6 1a 6b 104ab 1ka 96kb 16kab 0a 60b 88abka 6kb 104kab 8a 4b 00ab4ka 16kb 76kab 4a 7b 184ab 54ka 10kb 164kab ( 14a 4b Solutio for choice: 7 a 96b 16ab1ka 6kb 104kab 0a 60b 88ab1ka 96kb 16kab 58a 174b 0ab4ka 10kb 164kab 46a 18b 14ab4ka 16kb 76kab ( 14a 4b Solutio for choice: 14a 4b 48ab 0ka 90kb 48kab 16a 48b 1ab14ka 4kb 48kab 1a 9b 18ab5ka 156kb 96kab 17a 51b ab6ka 108kb 96kab Solutio for choice: 0a 90b 4ab 14ka 4kb 9kab 16a 48b 88ab0ka 90kb 4kab 5a 159b 8ab6ka 108kb 140kab Y ( 9a 117b 10ab 5ka 156kb 5kab ( 1a 9b Solutio for choice: 4 6a 18b 5ab 6ka 0kb 44kab 16a 48b 8ab 10ka 18kb 5kab ~ 5 ~ Solutio for choice: 8 0a 60b 88ab ka 96kb 16kab 1a 6b 104ab 0ka 60kb 88kab X ( 4a 7b 184ab 58ka 174kb 0kab 8a 4b 00ab46ka 18kb 14kab ( 14a 4b Solutio for choice: 9 a 96b 16ab 0ka 60kb 88kab 1a 6b 104abka 96kb 16kab X ( 54a 16b 76ab 46ka 18kb 14kab Y ( 4a 10b 164ab 58ka 174kb 0kab (.6 Patter: Rerite ( as u 14a 4b (k 1 ( v (11
(11 is ritte i the form of ratio as u (k 1 ( v, 0 v u (k 1 Which is equivalet to the system of double equatios 0 (1 0 Solvig the above system (1 by applyig the method of cross multiplicatio, e get u k 6k 6 v 4k (1 Usig (1 ad (, e get the correspodig o-zero iteger solutios of (1 to be, k 6k 8 4k, k 6 6k 4 4k X (, 4k 1k 14 4k, 4k 9 1k 10 4k (,.7 Properties a -1 a -1 6t4, a 4t 4SO a(1 60 4, a 8a (1 0 1 ( a1 t4, a(1 4 Pra (7-4ka-1 0 a - a a 5 a1 (a1 t4, a( 8 1(a k 1 4P a(5 0 a-1 ( a-1 t (11 4t (1 6k(4a-1 0.8 Remark 4, a 6, a (11 may also be expressed i the form of ratios i three more differet ays that are preseted belo: Way 1 u (k 1 ( v ( v u (k 1 Way u (k 1 ( v ( v u (k 1 Way u (k 1 ( v ( v u (k 1 ~ 6 ~ Solvig each of the above system of equatios by folloig the procedure as preseted i Patter:, the correspodig iteger solutios to (1 are foud to be as give belo: Solutio for ay 1,, 8 6k k 4k,, 6 4 6k k 4k,, 14 1k 4k 4k,, 9 10 1k 4k 4k (, Solutio for ay,, 6 4 k 6k 4k,, k 6k 8 4k,, 9 10 4k 1k 4k,, 14 4k 1k 4k (, Solutio for ay,, 6 6k k 4k,, 6k k 4k X (,, 9 1k 4k 4k,, 1k 4k 4k (,. Coclusio I this paper, e have preseted differet choices of iteger solutios to the homogeeous biquadratic equatio ith five ukos (x - y (x y (k k ( X Y. It is orth metioig here that i (, the liear trasformatios for X ad Y may also be cosidered as I X=uv+, Y=uv- ad II X=uv+1, Y=uv-1. Employig the above to forms of trasformatios for X,Y differet values for X ad Y are obtaied. To coclude, as biquadratic equatios are rich i variety, oe may cosider other forms of biquadratic equatios ad search for correspodig properties. 4. Ackoledgemet The fiical support from the UGC, Ne Delhi (F.MRP- 51/14(SERO/UGC dated march 014 for a part of this ork is gratefully ackoledged. 5. Refereces 1. L.E. Dickso, History of Theory of Numbers, Vol.11, Chelsea Publishig Compay, Ne York (195.. L.J. Mordell, Diophatie equatios, Academic Press, Lodo (9.. Carmichael, R.D., The theory of umbers ad Diophatie Aalysis, Dover Publicatios, Ne York (1959
4. Telag, S.G., Number theory, Tata McGra Hill publishig compay, Ne Delhi (1996 5. M.A. Gopala & P. Shamugaadham, O the 4 4 4 4 biquadratic equatio x y z, Impact J.Sci tech; vol.4, No, 4,111-115, (010. 6. M.A. Gopala & G. Sageeth Itegral Solutios of No-Homogeeous Quartic equatio 4 4 x y ( 1( z, Impact J.Sci Tech; Vol.4 No., 15-1, 010. 7. M.A. Gopala & Padm Itegral Solutio of No- 4 4 Homogeeous Quartic equatio x y z, Atarctica J. Math., 7(4,71-77,010. 8. M.A. Gopala & V. Padichelvi o the solutios of the 4 Biquadratic Equatio ( x y ( z 1 Paper preseted i the iteratioal coferece o Mceathematical Methods ad Computatio, Jamal Mohamed college, Tiruchirapalli, July 4-5, 009 9. M.A.Gopala, S. Vidhyalakshmi ad K. Lakshmi, O The Biquadratic Equatio ith Four ukos x xy y ( z z IJPAM (RIP vol.5, No.1, 7-77, 01 10. M.A. Gopala & J. Kaligarai, Quartic equato i five 4 4 ukos x y ( z p Bulleti of Pure ad Applied Scieces Vol8E, No., 05-11, 009 11. M.A. Gopala, J. Kaligarai, Quartic equato i five 4 4 ukos x y ( z p. Impact J.Sci Tech; Vol.4 No., 15-1, 009 1. M.A. Gopala, S. Vidhyalakshmi, A.Kavith observatios o the biquadratic ith five ukos 4 4 x y x x y Z( X Y IJESM, Vol, Issue, 19-00, Jue 01 1. M.A. Gopala, S. Vidhyalakshmi, E. Premalath o the homogeeous biquadratic ith five ukos 4 4 x y 8( z p IJSRP, Vol 4, issue 1, 1-5, Ja 014. ~ 7 ~