Interntionl Reserch Journl of Engineering nd echnology IRJE) e-issn: 395-56 Volume: Issue: 4 July-15 www.irjet.net p-issn: 395-7 ON HE ERNARY QUADRAIC DIOPHANINE EQUAION 4x y ) 3xy 16 M.A.Gopln 1 *, S.Vidhylkshmi, R.Presenn 3 J.Umvthy 4 1 Professor, Deprtment of Mthemtics, SIGC, richy-6, mil Ndu,Indi E-mil: myilgopln@gmil.com Lecturer, Deprtment of Mthemtics, SIGC, richy-6, mil Ndu,Indi E-mil: vidhysigc@gmil.com 3,4 Mphil Scholr, Deprtment of Mthemtics, SIGC, richy-6, mil Ndu, Indi E-mil:presennteddy9@gmil.com, umvthy99@gmil.com ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstrct - 4x y ) 3xy 16 1) he ternry homogeneous qudrtic eqution given by 4x y ) 3xy 16 representing is nlyed for its non-ero distinct integer solutions. A few interesting reltions between the solutions nd specil polygonl nd pyrmided numbers re presented. Key Words: ernry qudrtic, integer solutions, figurte numbers, homogeneous qudrtic, polygonl number, pyrmidl numbers. 1. INRODUCION: he Diophntine equtions offer n unlimited field for reserch due to their vriety [1-3].In prticulr, one my refer [4-] for qudrtic equtions with three unknowns. his communiction concerns with yet nother interesting eqution 4x y ) 3xy 16 representing homogeneous qudrtic eqution with three unknowns for determining its infinitely mny non-ero integrl points. Also, few interesting reltions mong the solutions re presented..mehod OF ANALYSIS: he ternry qudrtic diophntine eqution to be solved is o strt with, it is seen tht 1) is stisfied by the following integer triples x, y, ): 48,-16,8),,-16,8) However, we hve other choices of solutions to 1) which re illustrted below: he substitution of the liner trnsformtions x = u+v ; y = u-v u,v ) ) in 1) leds to ke 11v 16 3), 5 11b 4) where,b re non-ero distinct integers. Different ptterns of solutions of 1) re illustrted below PAERN: 1 Write 16) s 15, IRJE ISO 91:8 Certified Journl Pge 1936
Interntionl Reserch Journl of Engineering nd echnology IRJE) e-issn: 395-56 Volume: Issue: 4 July-15 www.irjet.net p-issn: 395-7 16 5 i * 5 i 5) 4) [ 6, ) 3, )] is perfect squre. 5) x,) 4pr 88mod88) Substituting 4),5) in 3) nd employing the method of fctorition, we ve * i 5 i 5 i i 5 i 5 i Equting the positive nd negtive fctors, we get i 5 i i 5 i 5 i 5 i 6) 7) Equting the rel nd imginry prts in 6) u u, 5 11b b v v, 5 11b 1b Substituting the vlues of u nd v in ), we gets x x, 1 b 1b 8) y y, 3b 9) hus 8),9) nd 4) represent non-ero distinct integrl solutions of 1) in two prmeters. A few interesting properties re s follows:- 1) x,1) 4pr mod 64) ) x,) 1pr 88mod34) 3) 11mod15) x,1) y,1),1) 15pr PAERN: Write 16) s 16 - * 5 i 5 i 1) Substituting 4),1) in 3) nd employing the method of fctorition, we ve * i 5 i 5 i i 5 i 5 i Equting the positive nd negtive fctors, we get i 5 i i 5 i 5 i 5 i Equting the rel nd imginry prts in u u, 5 11b b v v, 5 11b 1b Substituting the vlues of u nd v in ), we get 1) x x, 3b 13) y y, 1 b 1b 14) hus 13),14) nd 4) represent non-ero distinct integrl solutions of 1) in two prmeters. 15, IRJE ISO 91:8 Certified Journl Pge 1937
Interntionl Reserch Journl of Engineering nd echnology IRJE) e-issn: 395-56 Volume: Issue: 4 July-15 www.irjet.net p-issn: 395-7 A few interesting properties re s follows:- 1) y,3) 4pr 198mod ) y, pr 4mod 46) 3) 13mod1) b x,) y,), ) pr 4) [ 5 b, 4 b, ] is perfect squre. 5) y 3, ) pr 9mod58) PAERN:3 Substituting the liner trnsformtions u x - v x in 3), we get *) x 5 15) which is stisfied by pq 16) x 55p q 17) p 55q 18) From *), we get, u p 55q pq 19) v p 55q 1 pq ) Substituting the vlues of u nd v in ), we ve x xp,q) p 11q 1 pq 1) A few interesting properties re s follows:- 1) x, ) 11pr 8mod86) ) x,1) pr 11mod14) 3) 19mod 37) x,1) y,1),1) 57 pr 4) [ 1 q, q) 4 q, q)] is perfect squre. 5) x, ) y, ) 15t PAERN:4 Write eqution 15) s x) x) ) ) 3) he bove eqution is written in the form of rtio s x, x 4) he eqution 4) is equivlent to the following two equtions x 11 5) - x 6) Applying the method of cross multipliction, we get, x 11 5 5 11 herefore, X x, -11 5, - y yp,q) -3 pq ) hus 1) ) nd 17) represent non-ero distinct integrl solutions of 1) in two prmeters. =, 11 5 ) 7) Substituting in the vlues of X nd in *), we ve 15, IRJE ISO 91:8 Certified Journl Pge 1938
Interntionl Reserch Journl of Engineering nd echnology IRJE) e-issn: 395-56 Volume: Issue: 4 July-15 www.irjet.net p-issn: 395-7 u u v v, 11 5, 11 5 Substituting in the vlues of u nd v in ), we've x x, 1 8) y y, 9) hus 8),9) nd 7) represent non-ero distinct integrl solutions of 1) in two prmeters. A few interesting properties re s follows:- 1) 1, 1, 11 5t ) x,, 16t x 11 5 5 11 herefore, x x, 11 5, - =, 11 5 ) 34) Substituting the vlues of x nd in **), we ve u u, 11 5 v v, 11 5 1 3), ) 64t 4) x, -, 3t Substituting in the vlues of u nd v in ), we ve x x, 1 1 35) 5) x,, 15pr 44mod 9) y y, 3 36) PAERN: 5 Write eqution 15) s x) x) ) ) 3) he bove eqution is written in the form of rtio s x, x 31) he eqution 31) is equivlent to the following two eqution - x 11-3) - x 33) hus 35),36) nd 34) represent non-ero distinct integrl solutions of 1) in two prmeters. A few interesting properties re s follows:- 1) x,, 4t ) x 3, 1pr 198mod 46) 3) [ 6,6 )] is perfect squre. 4) x,1) y,1),1) 11pr 15mod 33) 5),1),1) 11t 5 3 Applying the method of cross multipliction, we get, 15, IRJE ISO 91:8 Certified Journl Pge 1939
Interntionl Reserch Journl of Engineering nd echnology IRJE) e-issn: 395-56 Volume: Issue: 4 July-15 www.irjet.net p-issn: 395-7 PAERN: 6 Write eqution 15) s x) x) 5 ) ) 37) he bove eqution is written in the form of rtio s x 5, x 38) he eqution 38) is equivlent to the following two equtions x 5 39) x 4) Applying the method of cross multipliction, we get, x 55 55 herefore, x x, 55 A few interesting properties re s follows:- 1) x, y, 18t ) x, 6) 11pr 7mod 3) 3) [,6] is perfect squre. 4) 3mod18) x,1) y,1),1) pr 5) 1, 1, t 55 REMARK: In ddition to 4),31),38) nd 15) my lso be expressed in the form of rtios in four different wys tht re presented below: WAY1: x WAY: x, x x =, 55 41) WAY3: Substituting the vlues of x nd in **), we ve u u, 55 v v, 55 1 Substituting in the vlues of u nd v in ), we ve x Wy4: x 5 x 55 x x x, 11 1 4) y y, 3 43) hus 4),43) nd 41) represent non-ero distinct integrl solutions of 1) in two prmeters. Solving ech of the bove system of equtions by following the procedure presented in pttern 4),5),6), the corresponding integer solutions to 1)re found tobe s given below: Solution for wy 1: x x, 1 1 y y, 3 15, IRJE ISO 91:8 Certified Journl Pge 194
Interntionl Reserch Journl of Engineering nd echnology IRJE) e-issn: 395-56 Volume: Issue: 4 July-15 www.irjet.net p-issn: 395-7 =, 5 11 Solution for wy : x x, 1 1 y y, 3 =, 5 11 ) Solution for wy 3: x x, 11 1 y y, 3 =, 55 Solution for wy 4: x x, 11 1 y y, 3 =, 55 3.CONCLUSION: In this pper, we hve obtined infinitely mny non-ero distinct integer solutions to the ternry qudrtic diophntine eqution represented by 4x y ) 3xy 16 As qudrtic equtions re rich in vriety, one my serch for other choices of qudrtic eqution with vribles greter thn or equl to 3 nd determine their properties through specil numbers. ACKNOWLEDGEMEN he finncil support from the UGC, New Delhi F.MRP-51/14 SERO/UGC) dted Mrch 14) for prt of this work is grtefully cknowledged. REFERENCES 1. Dickson LE.History of heory of Numbers nd Diophntine Anlysis,Vol,Dove publictions,new York 5.. Mordell LJ. Diophntine Equtions Acdemic Press, Newyork 197. 3. Crmichel RD.he heory of Numbers nd Diophntine Anlysis, Dover publictions,,newyork 1959. 4. Gopln MA,Geeth D. Lttice points on the Hyperboloid of two sheets x 6xy y 6x y 5 4 Impct J Sci ech 1;4:3-3. 5. Gopln MA,Vidhylkshmi S,Kvith A,Integrl points on the Homogeneous x -7y 1;1):17-136.,he Diophntus J Mth 6. Gopln MA,Vidhylkshmi S, Sumthi G.Lttice points on the Hyperboloid of one sheet 4 x 3y 4. Diophntus J Mth 1; 1): 19-115. 7. Gopln MA,Vidhylkshmi S,Lkshmi K. Integrl points on the Hyperboloid of two sheets 3y 7x - 1. Diophntus J Mth 1; 1): 99-17. 8. Gopln MA,Vidhylkshmi S, Mllik S.Observtions on Hyperboloid of one sheet x y 1-6.. Bessel J Mth 1; 3): 9. Gopln MA,Vidhylkshmi S,Ush Rni R, Mllik S,Integrl points on the Homogeneous 6 3y 1;61):7-13. x,he Impct J Sci ech 1. Gopln MA,Vidhylkshmi S,Sumthi G,Lttice points on the Elliptic prbolid 9x 4y Mthemtics 1;m74):379-385,Advnces in heoreticl nd Applied 11. Gopln MA,Vidhylkshmi S,Ush Rni R,Integrl points on the non- homogeneous 4xy 8x 4, Globl Journl of Mthmtics nd Mthmticl sciences 1;1):61-67 15, IRJE ISO 91:8 Certified Journl Pge 1941
Interntionl Reserch Journl of Engineering nd echnology IRJE) e-issn: 395-56 Volume: Issue: 4 July-15 www.irjet.net p-issn: 395-7 1. Gopln MA,Vidhylkshmi S,Lkshmi K.,Lttice points on the Elliptic prboloid 16y 9 4x, Bessel J of Mth 13; 3): 137-145. 13. Gopln MA,Vidhylkshmi S,Um Rni J, Integrl points on the Homogeneous 4y x 37,Cyley J of Mth 13;):11-17. 14. Gopln MA,Vidhylkshmi S, Kvith A. Observtions on the Hyperboloid of two sheet 7x 3y y x) 4. Interntionl Journl of Ltest Reserch in Science nd technology 13; ): 84-86. 15. Gopln MA,Sivgmi B. Integrl points on the homogeneous 3x Mthemtics 13; 84): 4-9. 6y. ISOR Journl of 16. Gopln MA,Geeth V. Lttice points on the homogeneous x journl of Science 13; : 93-96. 8y 6xy. Indin 17. Gopln MA, Vidhylkshmi S,Mheswri D. Integrl points on the homogeneous 35 14; 7: 6-1. x 3y. Indin journl of Science 18. Gopln MA, Vidhylkshmi S,Umrni J. On the ernry Qudrtic Diophntine Eqution 6x y ) 8xy 1. Sch J Eng ech 14; A): 18-11. 19. Meen K,Vidhylkshmi S, Gopln MA, Priy IK. Integrl points on the 3x y ) -5xy 47. Bulletin of Mthemtics nd sttistic Reserch 14; 1): 65-7.. Gopln MA, Vidhylkshmi S,Niveth S.On ernry Qudrtic Diophntine Eqution 4x y ) 7xy 31. Diophntus J Mth 14; 31): 1-7. 1. Meen K,Vidhylkshmi S, Gopln MA,hngm SA. Integrl solutions on the homogeneous 8 4x 3y sttistic Reserch 14; 1): 47-53.. Bulletin of Mthemtics nd. Snthi J,Gopln MA, Vidhylkshmi. Lttice points on the homogeneous 8x y ) 15xy 56 Mth Stt 14; : 9-3.. Sch Journl of Phy 15, IRJE ISO 91:8 Certified Journl Pge 194