<s>[INST] Are there infinitely many positive integer solutions to the equation $\overline{x^2y^2} = z^2\,$? 
Do there exist infinitely many solutions to the equation $$\overline{x^2y^2} = z^2$$ where $x,y,z$ are positive integers? Note: the notation $\overline{xy}$ means a concatenation of the numbers $x$ and $y$ to form one number.

One solution to this equation is $(x,y,z) = (4,9,41)$. Then we have $\overline{4^29^2} = 1681 = 41^2$. If $y = \overline{a_1 \ldots a_n}$, then $\overline{x^2y^2} = 10^{2n} x^2+y^2 = z^2$, but I didn't see how to find infinitely many integer solutions to this.
 [/INST] $$\overline{2^2(3\cdot10^k)^2}=(7\cdot10^k)^2$$
and in general, if $(a,b,c)$ is a solution, then so is $(a,b\cdot10^k,c\cdot10^k)$
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