Homogeneous Bi-Quadratic Equation With Five Unknowns

Transcription

1 Interntionl Journl of Mthemtics Reserch. ISSN Volume 6, Number (01), pp Interntionl Reserch Publiction House Homogeneous Bi-Qudrtic Eqution With Five y p q Unknowns S.Vidhylkshmi, M.A.Gopln nd T.R.Ush Rni Deprtment of mthemtics,shrimthi Indir GndhiCollege,Trichy.6000 vidhysigc@gmil.com,myilgopln@gmil.com,ushrnisigc@gmil.com, ABSTRACT: We obtin infinitely mny non-ero integer quintuples stisfying the the Biqudrtic eqution with five unknowns.vrious interesting properties mong the vlues of, y,,p nd q re presented. KEYWORDS: Biqudrtic eqution with five unknowns, integrl solutions MSc 000 Mthemtics Subject Clssifiction: 11D5 NOTATIONS: n 1 m t( m, n) n1 nn 1 P m n [( m ) n 5 m] 6 1 ( OH ) ( n n n 1) n( n 1) P n INTRODUCTION The theory of Diophntine equtions offers rich vriety of fscinting problems. In prticulr, biqudrtic diophntine equtions, homogeneous nd non-homogeneous hve roused the interest of numerous mthemticins since ntiquity [1-5]. In this contet, one my refer [6-11] for vrious problems on the biqudrtic diophntine equtions with four vribles nd[ 1-15] for 5 vribles However, often we come cross non-homogeneous biqudrtic equtions nd s such one my require its Pper code: 178-IJMR

2 6 S.Vidhylkshmi, M.A.Gopln nd T.R.Ush Rni integrl solution in its most generl form. It is towrds this end, this pper concerns with the problem of determining non-trivil integrl solutions of the homogeneous eqution with five unknowns given by y p q A few reltions mong the solutions re presented. METHOD OF ANALYSIS: The homogeneous bi-qudrtic eqution with five unknowns to be solved is y p q (1) Introducing the liner trnsformtions = u + v, y = u - v, = v, p= u+ α, q= u- α () in (1),it is written s u v () Eqution () is solved through five different wys nd thus, in view of (), we obtin five different ptterns of non-ero distinct integer solutions to (1). PATTERN :1 Let b () Write s 1 i 1 i Using () & (5) in () nd employing the method of fctorition, define 1 i i u i v b Equting rel nd imginry prts in the bove eqution, we get u b b v b b Substituting the vlues of u, v in () we hve b b y 6 b 8 b 8 b (6) p b q b b Thus ()&(6) represent the non-ero distinct integer solutions of (1). (5)

3 Homogeneous Bi-Qudrtic Eqution 7 1 p t, 6, 5mod 9 y 1, 6 p n 0, 8t 10, t 50, mod 69 p, 1 q, 1 8 p t, q, 1 8t, t8, mod 11 PATTERN : Insted of (5), we write s 5 i 5 i is perfect squre. Using () & (7) in () nd employing the method of fctorition,define 5 i i b u i v Equting the rel nd imginry prts in the bove eqution,we get 1 u 5 10 b b 1 v b 10 b Replcing by A, b by B in the bove equtions& substituting u, v in(), we get (7) 18 A y 1 A 6 B B 18 AB AB 1 A B 10 AB p A 1 B 1 AB q 6 A 8 B 1 AB (8) Thus ()&(8) represent the non-ero distinct integer solutions of (1). A A 1, 18 p A t8, A 6(mod 89) y A, A 1 16OH A 96t t170, A (mod 8) 1, B t 50, B 1 (mod 97 ) p 1, B t 6, B 1(mod ), A q (A, A ) 1 t 10,A t 98, A 5 (mod 97 )

4 8 S.Vidhylkshmi, M.A.Gopln nd T.R.Ush Rni PATTERN : Write () s v u (9) Assume v b (10) Write s 1 1 (11) Using (10) & (11) in (9)nd employing the method of fctorition,define 1 u b Equting the rtionl nd irrtionl prts in the bove eqution, we get b b u b 6 b Substituting the vlues of u, α in () we hve 6 6 b y b 6b 1 b (1) p 6 b 8 b q b Thus (9)&(1) represent the non-ero distinct integer solutions of (1). 5 (, 1) 6t 1 p 0 (, y ( 1, ) 6 p t (6, ) 8(mod 9) (,1) t ( 6, ) (mod 11) p ( 1, b) t( 6, b ) 6(mod 9) q ( 1, 1) p 0 ) PATTERN : () is written in the form of rtio s u ( v), b 0 v u b (1) which is equivlent to the system of equtions bu v b (1) 0

5 Homogeneous Bi-Qudrtic Eqution 9 b 0 u bv (15) The bove system is stisfied by u b b v b b (16) b Substituting (16) in (),the corresponding non- ero distinct integrl solutions of (1) re given by 6b y b b 8b p b 8b q b b (, 1) 18 OH 0 5 y (, 1) t(10, ) p t (mod 11),, 1 8p t10, 8mod19 p (1, b) q 1, b t 10, (mod 5) q ( 1, ) p t0, 16mod 19 PATTERN : 5 Also,() is written in the form of rtio s u v, b 0 ( v) u b (17) which is equivlent to the system of equtions bu v b 0 (18) u bv b (19) 0 The bove system is stisfied by u b b v b b (0) b

6 50 S.Vidhylkshmi, M.A.Gopln nd T.R.Ush Rni Substituting (0) in (),the corresponding non- ero distinct integrl solutions of (1) re given by b b y 6b 8 b 8b p b q b b, p t 1 6, 5(mod 9) y (, 1) 18OH 0 5 (, 1) 8t 16 p t10, (mod 11) (, 5 p( 1, ) 8p t 10, mod 11 q (, ) t10, t 66, 1mod 17 ) Conclusion In this pper,we hve presented different ptterns of non-ero distinct integer solutions to the homogeneous biqudrtic eqution with 5 unknowns y p q.as the biqudrtic diophntine equtions re rich in vreity due to the definition of diophntine equtions, One my consider biqudrtic diophntie equtions with unknowns 5 nd serch for their integer solutions long with the properties. REFERENCES: [1] L.E. Dickson, History of Theory of Numbers, Vol.11, Chelse publishing compny, New York (195) [] L.J. Mordell, Diophntine equtions, Acdemic press, London (1969) [] Crmichel, R.D., The theory of numbers nd Diophntine Anlysis, Dover publictions, New York (1959) [] Telng, S.G., Number theory, Tt Mc Grw Hill publishing compny, New Delhi (1996) [5] Nigel, D.Smrt, The Algorithmic Resolutions of Diophntine Equtions, Cmbridge University press, London (1999) [6] Gopln, M.A., & V. Pndichelvi On the Solutions of the Biqudrtic eqution [( y )] ( 1) w pper presented in the interntionl conference on Mthemticl Methods nd Computtion, Jml Mohmmed College, Tiruchirpplli, July -5 (009).

7 Homogeneous Bi-Qudrtic Eqution 51 [7] Gopln, M.A., & P. Shnmugnndhm, On the biqudrtic eqution y w, Impct J.Sci tech; Vol., No., , (010). [8] Gopln, M.A., & G. Sngeeth, Integrl solutions of Non-homogeneous Qurtic eqution y ( 1)( w ), Impct J.Sci Tech; Vol. No., 15-1, (010) [9] Gopln, M.A., & R. Pdm, Integrl solution of Non-homogeneous Qurtic eqution ( y ) w, Antrctic J. Mth., 7(), 71-77, (010) [10] Gopln, M.A., nd P. Shnmugnndhm On the Biqudrtic Eqution y ( y) ( k n ) w Bessel J.Mth., (), 87-91(01). [11]Meen.K,Vidhylkshmi.S,Gopln.M.A,ArthyThngm.S On the Biqudrtic Eqution y 9w,IJOER,Vol.,issue 1, 57-60,(01) [1] Gopln.M.A.,& Kligrni.J.Qurtic eqution in 5 unknowns y ( w ) p Bulletin of Pure nd Applied Sciences, Vol8(N0.),05-11,(009) [1] Gopln.M.A.,& Kligrni.J.Qurtic eqution in 5 unknowns y ( w ) p Bessel J. Mths., 1(1),9-57,(011) [1] Vidhylkshmi.S,Gopln.M.A.,Kvith.A Observtions on the biqudrtic eqution with 5 unknowns y y( y ) Z ( X Y ) ) IJESM,Vol,issue,19-00 June(01) [15] Gopln.M.A., Vidhylkshmi S Premlth.E., On the homogeneous biqudrtic eqution with 5 unknowns y 8( w ) p IJSRP,Vol,issue1, 1-5, Jn (01)

8 5 S.Vidhylkshmi, M.A.Gopln nd T.R.Ush Rni