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I was just thinking about squares while randomly punched numbers into my calculator and I was wondering do there exist infinitely many squares such that $\sqrt{\sqrt{a}}$ is a square and $a$ is also a square? For example:

$$\begin{array}{l|c} a & \sqrt{\sqrt{a}} \\ \hline 256 & 4 \\ 16 & 2 \end{array}$$

I tried some numbers and it seemed to work for a couple but I want to prove that there are infinitely many of these. I am not well versed in proving (at all) so if someone could provide me with preliminary steps then that would be great!


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Hint: Let $a=n^8, n=0,1,2...$. – user60725 Apr 30 '13 at 13:30
@BarackObama That was a very helpful hint! – Jeel Shah Apr 30 '13 at 17:23
up vote 13 down vote accepted

If $\sqrt{\sqrt a}$ is a square itself then we can write it as $n^2$, therefore $a=\left(\left(n^2\right)^2\right)^2=n^8$.

From this it is easy to conclude the infinitude of these numbers.

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