### AIMS Mathematics

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

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

• 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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###### 通讯作者: 陈斌, bchen63@163.com
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沈阳化工大学材料科学与工程学院 沈阳 110142

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