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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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  • © 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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