On gaussian diophantine quadruples

  • Authors

    • S. Vidhyalakshmi
    • M. A.Gopalan
    • S. Aarthy Thangam
    https://doi.org/10.14419/ijet.v7i4.29678
  • Diophantine Quadruple, Double Diophantine Equations, Gaussian Diophantine Quadruples, Integer Solutions, Pell Equations.

  • This paper concerns with the problem of constructing gaussian diophantine quadruples with the property that the product of any two distinct gaussian integers added with 1 and 4 in turn is a perfect square. The construction of gaussian diophantine quadruple  is illustrated through employing the non-zero distinct integer solutions of the system of double diophantine equations. The repetition of the above process leads to the generation of sequences of gaussian diophantine quadruples with the given property.

     

     

  • References

    1. [1] L.E. Dickson, History of the theory of numbers, Chelsea Publishing Company, Newyork, Vol.2, 1966, 513-520.

      [2] E. Brown, Sets in which is always a square, Math. Comp., 45, (1985), 613-620.https://doi.org/10.2307/2008150.

      [3] J. Morgado, Generalization of a result of Hoggatt and Bergum on Fibonacci numbers Portugaliae Math, 42, (1983-1984), 441-445.

      [4] J. Morgado, note on a Shannon’s theorem concerning the Fibonacci numbers and Diophantine quadruples, Portugaliae Math, 48, (1991), 429-439.

      [5] G. Srividhya, Diophantine quadruples for Fibonacci numbers with property , Indian Journal of Mathematics and Mathematical Sciences, 5, (2009), 57-59.

      [6] M.A. Gopalan, G. Srividhya, Diophantine quadruples for Fibonacci and Lucas numbers with property , Diophantus J. Math., 1, (2012), 15-18.

      [7] A. Dujella, Generalization of a problem of Diophantus, Acta Arith, 65, (1993), 15-27.https://doi.org/10.4064/aa-65-1-15-27.

      [8] A. Dujella, A problem of Diophantus and pell numbers, Application of Fibonacci Numbers, Vol.7, (G.E. Bergum, A.N. Philippou, A.F. Horadan eds.) Khowet Dordrecht, (1998), 61-68.https://doi.org/10.1007/978-94-011-5020-0_9.

      [9] Y. Fujita, The extensibility of Diophantine pairs , J. Number Theory, 128, (2008), 322-353.

      [10] Y. Fujita, A. Togbe, Uniqueness of the extension of the -triple , Notes Number Theory, Discrete Math, 17, (2011), 42-49.

      [11] Z. Cerin, on extended Euler quadruples, AttiSemin Mat Fis. Univ Modena Reggio Emilia 58, (2011), 101-120.

      [12] Y. Zhang, Diophantine triples and extendibility of [1, 2, 5] and [1, 5, 10], Master Thesis, Central Michigan University, 2011.

      [13] Lj. Bacic, A. Filipin, The extendibility of D(4)-pairs Math, Commun, 18, (2013), 447-456.

      [14] K. Meena, S. Vidhyalakshmi, M.A. Gopalan, G. Akila, R. Presenna, Formation of special diophantine quadruples with property , The international journal of Science and Technology 2, (2014) , 11-14.https://doi.org/10.9790/5728-10321214.

      [15] Andrej Dujella and Florian Luca, Diophantine m-tuples for primes Intern. Math.Research, Notes 47, (2005), 2913-2940.https://doi.org/10.1155/IMRN.2005.2913.

      [16] M.A. Gopalan, K. Lakshmi, S. Vidhyalakshmi, Gaussian Diophantine quadruples with property D(1), IOSR, Vol 10(3), version II, (2014), 12-14.https://doi.org/10.9790/5728-10321214.

      [17] M.A. Gopalan, S. Mallika, S. Vidhyalakshmi, On the extendibility of the Gaussian 2-tuple to Gaussian Diophantine quadruple with property D(1), International Journal of Innovation in Science and Mathematics, 2(3), (2014), 310-311.https://doi.org/10.9790/5728-10321214.

      [18] M.A. Gopalan, S. Mallika, S. Vidhyalakshmi, On Diophantine quadruple with property where P is prime and , International Journal of Current Research, 6(5), (2014), 6810-6813.

      [19] K. Meena, S. Vidhyalakshmi, M.A. Gopalan, R. Presenna, Sequences of special dio-triples, IJMTT, 10(1), (2014), 43-46.https://doi.org/10.14445/22315373/IJMTT-V10P508.

      [20] M.A. Gopalan, S. Mallika, S. Vidhyalakshmi, On sequences of irrational diophantine quadruples, International Journal of Mathematical Sciences and Applications, 4(1), (2014), 128-135.

      [21] M.A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi, On strong rational diophantine quadruples with equal members, SJMPS, 1(2), (2014), 88-95.

      [22] M.A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi, On strong diophantine quadruples with property D(1), The IJST, 2(7), (2014), 17-22.https://doi.org/10.14445/22315373/IJMTT-V20P514.

      [23] M.A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi, “A special diophantine quadruple with property D (17)â€, IJMTT, 20(2), (April 2015), 110-112.https://doi.org/10.14445/22315373/IJMTT-V20P514.

      [24] M.A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi, “Construction of irrational gaussian diophantine quadruplesâ€, IJETMR, 1(1), (April 2015), 1-7.

      [25] P. Saranya and G. Janaki, “Construction of gaussian diophantine triples with the property D(25)â€, International Journal of Statistics and Applied Mathematics, 2(6), (2017), 301-302.

  • Downloads

  • How to Cite

    Vidhyalakshmi, S., A.Gopalan, M., & Aarthy Thangam, S. (2018). On gaussian diophantine quadruples. International Journal of Engineering & Technology, 7(4), 6947-6950. https://doi.org/10.14419/ijet.v7i4.29678