<s>[INST] why $x^2 - 25y^2 = 1$ cannot have integer solutions other than $( \pm 1,0)$ 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 : $x^2-25y^2=1$
 [/INST] 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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