Show that $x^4-20200y^2=1$ has no solution in postive integers Show that $x^4-20200y^2=1$ has no solution in postive integers.
This topic is a question for the Chinese middle school students' mathematics competition today, so I think this problem has a simple solution.

The Pell equation $z^2-20200y^2=1$ has the fundamental solution:
$$
x^2=z = 30729653461384352052499,\;
y = 216213087258427446135
$$
But $z$ is not a perfect square. All remaining solutions are $(z_m,y_m)$ with $z_m+y_m\sqrt{20200}=(z+y\sqrt{20200})^m$. Then we have to show that $z_m$ is not a perfect square for all $m$. I suppose that this is not the idea for the middle school, but at least one has a solution.
