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pell equations must have n to be nonsquared,so $x^2 - 25y^2 = 1$ is not a pell equation, but the thing is how can i show $x^2 - 25y^2 = 1$ have no integer solutions? or it has integer solutions.

Solve : $x^2-25y^2=1$

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    $\begingroup$ Factor the LHS (LHS = Left-Hand-Side). $\endgroup$
    – quasi
    May 12, 2017 at 6:38
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    $\begingroup$ $x=1$ and $y=0$ are integers. $\endgroup$ May 12, 2017 at 6:39
  • $\begingroup$ the LHS can be odd just take x odd and y even. $\endgroup$ May 12, 2017 at 6:43

1 Answer 1

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Given $x^2-25y^2=1$

$(x-5y)(x+5y)=1$
$1=-1 \cdot -1$ and $1=1 \cdot 1$ are the only ways to write $1$ as the product of two integers.
Hence we must have $x-5y=x+5y=1$ or $x-5y=x+5y=-1$ and so it gives the only solutions as $x= \pm 1$ and $y=0$ .

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  • $\begingroup$ Ha! And here my first impulse was to use modular arithmetic. Too much knowledge can make you look for too complicated answers, sometimes. $\endgroup$ May 12, 2017 at 6:45
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    $\begingroup$ Modular arithmetic is fine while proving that you have no solutions. Once you have one solution, it is all but useless. $\endgroup$ May 12, 2017 at 7:06

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