Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org Oservtion on the Bi-qudrtic Eqution ith Five Unnons 1 1 R.Anuselvi 1 nd N.Ahil * 1 Deprtment of Mthemtics A.D.M. College for Women Autonomous Ngpttinm 00 001 Tmil Ndu Indi. Deprtment of Mthemtics Thiru.Vi.K. Govt. Arts College Tiruvrur- 1000 Tmil Ndu Indi Astrct: In this pper e find sustntill mn non-ero numerl quintuples pqdet fulfilling the iqudrtic eqution ith five unnons 1 1 different ttrctive delings mong the solutions nd out of the ordinr numers octhedrl numers centered multilterl & prmidl numers re ehiited. Ke ords: Bi-Qudrtic eqution ith five unnons essentil solutions multilterl numer Prmidl numers Centered multilterl. 010 Mthemtics suject clssifiction: 11D5. Nottions Used: - Polgonl numer of rn n ith sie m. n Pm - Prmidl numer of rn n ith sie m. - Pronic numer of rn n. - Octhedrl numer of rn n. 1. INTRODUCTION Bi-qudrtic Diophntine Equtions uniform nd ununiform hve roused the ttention of mn Mthemticins since uncertint s cn e seen from [117-19]. In the cground one m refer [-1] for vrieties of prolems on the Diophntine equtions ith to three nd four vriles. This communiqué concerns ith the prolems of determining non-ero fundmentl solutions of i-qudrtic eqution in the si unnons represented 1 1. solutions nd prticulr multilterl numers re offered.. METHOD OF ANALSIS The Diophntine eqution representing the i-qudrtic eqution ith si unnons elo deliertion is 1 1 Introducing the liner trnsformtions A smll numer of ttrctive reltions eteen the u v u v u v u v in 1 it simplifies to 1 50
Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org u 1v 1 1 The ove eqution is solved through diverse methods nd thus one otins different sets of integer solutions to eqution 1. Set.1 Let 1 Sustituting in nd using method of fctorition define u i v 1 i i 5 Equting rel nd imginr prts e hve u v 1 1 Sustituting the vlues of u nd v in eqution the non-ero distinct integrl solutions of 1 re given 1 1 1 1 1 8 8 7 Properties: i ii 1 1 11t 8t mod 8 1 1 11t 1 8Pr 0 iii 1 1 1 OH 8t 7t 1 0 iv 1 1 t t SO 1 t 1t 1SO 1 0 v 1 1 t t 7t 1 t 8 1 mod9 vi 1 1 8t 19t5 vii 8 mod 88 t10 t 9 0 Note: In 5 replce 1 i 1 i 505
Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org u i v 1 i i 8 Folloing the process offered in set. 1 diverse solution is given 1 1 1 1 1 8 8 9 Set. cn e ritten s u 1v 1 1 1 10 Write 1 s 1 i i 1 11 Using 11 in 10 nd emploing the method of fctorition delinete i u i v 1 i i 1 Equting rel nd imginr prts & replcing nd e hve u 8 v 8 9 8 8 57 9 19 9 1 9 1 Using 1 & e get the integrl solutions of 1 to e 1 0 1 1 78 88 10 19 19 18 0 1 8 105 18 8 17 80 08 88 1 Properties: i 1 1 19t 8t 1SO 1 17t 1t ii 1 1 5t 78 Pr 1mod 1SO 0 50
Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org iii 1 1 t OH 8t 7t OH 0 iv 1 1 1 Ol 11t 1 Ol t 0 5 v 1 1 11P t 1t t 1t 1 19 mod 8 vi 1 vii 1 1 1 1 18t viii 7 5 7 5 587t Note : 1 1 8t GnO t 9 7t 1 0 7 1 1t 171t t 1 88mod8 9 50 900 900mod 880 i In 1 replce i u i v 1 i i i 15 Folloing the process offered in set. diverse solution is given 1 1 0 1 78 19 88 10 19 18 1 0 8 18 105 8 08 88 17 80 1 Cse 1: u 1v 1 1 1 Insted of 11 rite 1 s 1 i 1 i 1 c 17 9 Using 17 in 10 nd emploing the method of fctorition define 1 i u i v 1 i i 18 7 Equting rel nd imginr prts nd replcing 7 nd 7 507
Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org u 7 v 1 18 7 8 18 01 8 18 1 19 Using 19 & e get the integrl solutions of 1 to e 5 1 8 9 588 18 9 0 11 5 198 8 7 18 98 08 50 80 88 0 Note 1 i In 18 replce 7 1 i 7 u i v 1 i i Folloing the procedure presented in cse 1 different solution is given 1 5 8 9 18 588 15 11 9 5 0 18 8 1 7 i 188 98 50 08 80 58 7 1 Cse : u 1v 1 1 1 As n lterntive of 17 rite 1 s 8 i8 1 8 i8 5 Using 17 in 10 nd emploing the method of fctorition define 8 i8 u i v 1 i i 1 Equting rel nd imginr prts nd replcing 1 nd 1 508
Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org u 18 v 78 18 15 78 91 15 15 07 5 15 5 Using 5 & e get the integrl solutions of 1 to e 5 10 0 19 5 51 188 108 1 07 07 80 0 1 08 189 1998 10 1 81 780 8 08 Note 8 i8 In replce 1 8 i8 1 u i v 1 i i Folloing the prllel process s in cse the equivlent numerl solutions of 1 re given 10 5 19 0 5 188 51 07 1 108 08 07 0 80 8 8 1 i 1 1998 189 10 1 8 08 81 780 7 8 Set.III Rerite e get u 1 v 9 Choice: I u v v u 0 0 Using method of cross rtio e get u v 1 509
Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org Hence in oservtion of the prllel non-ero numerl solutions of 1 re 8 8 8 8 8 8 Properties i ii iii 1 1 t iv 1 1 1 1 1t 1 t 1 8t v 9t vi 5 5 8 t vii 8t viii 1 1 t 1t Pr t 7 8t t t 8t 10 9t 7OH 1t t 8 Pr 8 8SO SO 1Pr 8 0 1t 7 79t mod19 0 0 1t 0 10 1t 9 0mod9 90 7 0 In ddition to 0 9 m lso e uttered in the form of rtios in three diverse choices tht re otinle elo Choice: II u v v u Choice: III u v v u Choice: IV u v v u S1O Solving ech of the eceeding sstem of eqution the susequent process s offered in set.iii the prllel integer solution to 1 re estlish to e s given elo Solution for choice II: 0 510
Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org 511 8 8 8 8 8 8 Solution for choice III: 8 1 8 1 1 1 Solution for choice IV: 8 1 8 1 1 1. CONCLUSION: In this pper e hve offered distinct choices of integrl solutions to homogeneous iqudrtic eqution ith si unnons 1 1.To terminte s iqudrtic equtions re rich in multiplicit one m regrd s other forms of iqudrtic equtions nd rummge round for nlogous properties. REFERENCES [1] Crmichel R.D 1959 The theor of numers nd Diophntine nlsis Dover Pulictions Ne or. [] Dicson.I.E. 195 Histor of theor of numers Vol. Chelsi Pulishing Co. Ne or. [] Gopln.M.A. nd Anuselvi.R. 009 Integrl solutions of inr qurtic eqution reflections des ERA- JMS Vol.Issue pp 71-80 [] Gopln.M.A. nd Jni.G. 009 Oservtion on Act cienci Indic Vol.VM No.5 [5] Gopln.M.A. Vidhlshmi.S nd Devil.S 010 Ternr Qurtic Diophntine Eqution n [] Gopln M.A. Vijsnr.A nd Mnju Somnth 010 Integrl solutions of Impct journl of Science nd Technolog Vol. No. 19-157. [7] Gopln.M.A.nd Shnmugnndhm.P. 010 On the Biqudrtic eqution Impct journl of Science nd Technolog Vol. No 111-115. [8] Gopln M.A. Sngeeth.G.011 Integrl solutions of ternr non-homogeneous iqudrtic eqution Act cienci indic Vol.VII M No.799-80. [9] Gopln.M.A..Vidhlshmi.S. Sumthi.G01 On the ternr iqudrtic non-homogeneous eqution 1 Indin journl of EngineeringVol1 No.1. [10] Gopln.M.A..Vidhlshmi.S. Sumthi.G01 Integrl solutions of ternr iqudrtic non-homogeneous eqution
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