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AIMS Mathematics

2019, Volume 4, Issue 4: 1170-1180. doi: 10.3934/math.2019.4.1170
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Research article

On the Diophantine equations $x^2-Dy^2=-1$ and $x^2-Dy^2=4$

  • School of Mathematics and Information, China West Normal University, Nanchong 637009, P. R. China
  • Received: 03 April 2019 Accepted: 14 July 2019 Published: 19 August 2019

  • MSC : 11D25, 11B39

  • In this paper, using only the St$ \ddot{o} $rmer theorem and its generalizations on Pell's equation and fundamental properties of Lehmer sequence and the associated Lehmer sequence, we discuss the Diophantine equations $x^2-Dy^2 = -1$ and $x^2-Dy^2 = 4$. We obtain the relation between a positive integer solution (x, y) of the Diophantine equation $x^2-Dy^2 = -1$ and its fundamental solution if there is exactly one or two prime divisors of y not dividing D. We also obtain the relation between a positive integer solution (x, y) of the Diophantine equation $x^2-Dy^2 = 4$ and its minimal positive solution if there is exactly two prime divisors of y not dividing D.

    Citation: Bingzhou Chen, Jiagui Luo. On the Diophantine equations $x^2-Dy^2=-1$ and $x^2-Dy^2=4$[J]. AIMS Mathematics, 2019, 4(4): 1170-1180. doi: 10.3934/math.2019.4.1170

    Related Papers:

  • In this paper, using only the St$ \ddot{o} $rmer theorem and its generalizations on Pell's equation and fundamental properties of Lehmer sequence and the associated Lehmer sequence, we discuss the Diophantine equations $x^2-Dy^2 = -1$ and $x^2-Dy^2 = 4$. We obtain the relation between a positive integer solution (x, y) of the Diophantine equation $x^2-Dy^2 = -1$ and its fundamental solution if there is exactly one or two prime divisors of y not dividing D. We also obtain the relation between a positive integer solution (x, y) of the Diophantine equation $x^2-Dy^2 = 4$ and its minimal positive solution if there is exactly two prime divisors of y not dividing D.


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    [1] L. E. Dickson, History of the Theory of Numbers, Vol. II, Washington, Carnegie Institution of Washington, 1920.
    [2] D. H. Lehmer, An extended theory of Lucas' functions, Ann. Math., 31 (1930), 419-448.
    [3] J. G. Luo, Extensions and applications on störmer theory, Journal of Sichuan University, 28 (1991), 469-474.
    [4] J. G. Luo, P. Z. Yuan, On the solutions of a system of two Diophantine equations, Science China Mathematics, 57 (2014), 1401-1418.
    [5] J. G. Luo, A. Togbe, P. Z. Yuan, On some equations related to Ma's conjecture, Integers, 11 (2011), 683-694.
    [6] J. G. Luo, P. Z. Yuan, Square-classes in Lehmer sequences having odd parameters and their applications, Acta Arith., 127 (2007), 49-62.
    [7] H. Mei, L. Mei, Q. fan, et al. Extensions of störmer theorem, Journal of Yuzhou University, 12 (1995), 25-27.
    [8] P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989.
    [9] J. G. Luo, On the Diophantine equation $\frac{ax^m\pm 1}{ax\pm 1}=y^n$ and $\frac{ax^m\pm 1}{ax\pm 1}=y^n+1$, Chinese Annals of Mathematics, Series A, 25 (2004), 805-808.
    [10] Q. Sun, P. Z. Yuan, On the Diophantine equatins $(ax^n-1)/(ax-1)=y^2$ and $(ax^n+1)/(ax+1)=y^2$, Journal of Sichuan University, 26 (1989), 20-24.
    [11] P. Z. Yuan, A note on the divisibility of the generalized Lucas sequences, Fibonacci Quarterly, 40 (2002), 153-156.
    [12] P. Mihäilescu, Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math., 572 (2004), 167-196.
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  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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