International Journal of Innovative Research in Science, Engineering and Technology

International Journal of Innovative Research in Science, Engineering and Technology

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Construction of Strong Diophantine Quadruples and Pell Equation

M.A.Gopalan1 , S. Vidhyalakshmi2, E.Premalatha3
  1. Professor, Department of Mathematics, SIGC, Trichy, Tamilnadu, India
  2. Assistant Professor, Department of Mathematics, National College, Trichy, Tamilnadu, India
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Abstract

This paper aims at constructing strong Diophantine quadruples by employing the non zero distinct integer solutions of the Pell equations y2  Dx2 1 and y 2  Dx2  4 .

Keywords
Strong Diophantine Quadruples, Pellian equation, integer solutions.

INTRODUCTION
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IV. ACKNOWLEDGEMENT
*The finicial support from the UGC, New Delhi (F-MRP-5123/14(SERO/UGC) dated march 2014) for a part of this work is gratefully acknowledged.

References
  1. A.Baker,H.Davenport,The equations 3x2 2 y2 and 8x2 7 z 2 , Quart.J. Math.Oxford ser.20(2) 129-137 ,(1969).
  2. V.E.Hoggatt,G.E.Bergum, A Problem of Fermat and the Fibonacci sequence, Fibonacci Quart, 15, 323-330,(1977).
  3. A.F.Horadam, Generalization of a result of Morgudo,Portugaliae Math., 44, (1987),131-136.
  4. B.W.Jones, Asecond variation on a problem of diophantus and davenport, Fibonacci Quart,16, 155-165, (1978).
  5. C.Long, G.E.Bergum, On a problem of Diophantus in: Application of Fibonacci numbers, (A.N.Philippou,A.F.Horadam
  6. and .E.Bergum,eds),Kluwer,Dordrecht, Vol.2, 183-191 ,(1988).
  7. J.Morgado, Generalization of a result of Hoggatt and bergum on Fibonacci numbers , Portugaliae Math., 42, 441-445 ,(1983-1984).
  8. J.Morgado, “Note on a Shannon’s theorem concerning the Fibonacci numbers” Portugaliae Math., 48 , 429- 439,(1991).
  9. J.Morgado, “Note on the Cheyshev polynomials of the second kind” , Portugaliae Math., 52,(1995),363-378.
  10. H.Gupta and K.Singh, “On k-triad Sequences”, Internet.J.Math.Sci., 5, 799-804 ,(1985).
  11. A.F.Beardon and M.N.Deshpande, “Diophantine triples”, The Mathematical Gazette, 86, 253-260, (2002).
  12. E.Brown,”Sets in which xy+k is always a square”, Math.Comp.45(1985), 613-620.
  13. M.N.Deshpande,”Families of Diophantine Triplets”,Bulletin of the Marathawada Mathematical Society, 4, ,19-21 (2003).
  14. Y.Bugeaud,A.Dujella and Mignotte,”On the family of Diophantine triples (k 1,k 1,16k 3 4k) “, Glasgow Math.J.49, 333-
  15. 344 ,(2007).
  16. Tao Liqun “On the property 1 P ”,Electronic Journal of combinatorial number theory 7, #A47 , (2007).
  17. Y.Fujita,”The extensibility of Diophantine pairs (k-1,k+1)”,J.number theory 128, 322-353, (2008).
  18. G.Srividhya,”Diophantine Quadruples for Fibbonacci numbers with property D(1)”,Indian Journal of Mathematics and sMathematical Science,Vol.5, No.2, 57-59 ,(December 2009).
  19. M.A.Gopalan ,V.Pandichelvi, “The Non Extendibility of the Diophantine Triple (4(2m 1)2 n2 ,4(2m 1)n 1,4(2m 1)4 n4 8(2m1)3 n3 ) , Impact.J.sci.Tech, 5(1),25-28, 2011.
  20. Yasutsugu Fujita,Alain Togbe, “Uniqueness of the extension of the D(4k 2 ) -triple (k 2 4,k 2 ,4k 2 4) ” NNTDM 17 , 4,42 49 ,(2011).
  21. Gopalan.M.A , G.Srividhya,”Some non extendable P5 sets “, Diophantus J.Math.,1(1), 19-22 (2012).
  22. Gopalan.M.A, G.Srividhya,” Two Special Diophantine Triples “, Diophantus J.Math.,1(1), 23-27.(2012).
  23. K.Meena, S.Vidhyalakshmi, M.A.Gopalan, R.Presenna, “Sequences of special Dio-triples”, International Journal of Mathematics Trends& Technology, Vol 10, no 1,43-46,Jun 2014.
  24. M.A.Gopalan,K.Geetha, Manju Somanath,” Special Dio-triples”, Bulletin of Society for Mathematical services & Standards,Vol 3, No 2,41- 45,2014.
  25. M.A.Gopalan,V.Geetha, S.Vidhyalakshmi,” Dio3-tuples for special numbers –I”, Bulletin of Society for Mathematical services& Standards,Vol 3, No 2,11-18,2014.
  26. M.A.Gopalan, V.Pandichelvi,”On the extendiability of the Diophantine triple involving Jacobsthal Numbers (J2n1, J 2n1 3,2J2n J 2n1 J2n1 3) ”,International journal of Mathematics and Applications, Vol 2, no 1-2,1-3,(2009).
  27. M.A.Gopalan, S.Mallika and S.Vidhyalakshmi ,”On Diophantine quadruple with property D( p2 ) where p is a prime and
  28. p2 1(mod6) ,International Journal of current Research , Vol 6, issue 05, 6810-6813,may 2014.
  29. Gopalan.M.A , G.Srividhya,” Diophantine Quadruple for Fionacci and Lucas Numbers with property D(4) “, Diophantus J.Math.,1(1) ,15- 18,(2012).
  30. Andrej Dujella,Zagreb,Croatia,” The problem of Diophantus and Davenport for Gaussian Integers”, Glas.Mat.Ser.III,32,1-10 ,(1997).
  31. S.Vidhyalakshmi,M.A.Gopalan and K.Lakshmi,”Gaussian Diophantine quadruples with property D(1)”, IOSR Journal of Mathematics, Vol 10, issue 3, Ver.II,12-14,(may-jun 2014).
  32. M.A.Gopalan, S.Vidhyalakshmi E.Premalatha and R.Presenna ,” On the extendibility of 2-tuple to 4-tuple withproperty D(4)”, Bulletin of Mathematical Sciences and Applications,Vol 3, issue 2, 100-104, (2014).
  33. M.A.Gopalan ,G.Sumathi , S.Vidhyalakshmi ,”Special Dio-quadruple involviong Jacobsthal and Jacobsthal lucas Numbers with property D(k 2 1) ”,IJMSEA, Vol 8, no 3, 221-225,2014.
  34. K.Meena, S.Vidhyalakshmi, M.A.Gopalan,G.Akila and R.Presenna,”Formation of special diophantine quadruples with property D(6kpq)2 ”The IJST,Vol 2,issue 2, ,11-14,2014.
  35. Andrej Dujella, Vinko petricevic,”Strong Diophantine triples”,Experimental Math,Vol 17,No 1, 83-89 , (2008).
  36. A.Dujella and N.Saradha,” Diophantine m-tuple with elements in arithmetic progressions”,Indagationes Mathematicae, Vol 25,issue 1, 131- 136,Jan 2014.