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How to find all naturals $n$ such that $\sqrt{1\smash{\underbrace{4\cdots4}_{n\text{ times}}}}$ is an integer?

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    Do you mean $\sqrt{1\underbrace{4\cdots4}_{n-times}}$ or perhaps$\sqrt{\underbrace{1414\cdots14}_{n-times}}$?2012-07-14
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    I mean The first one .2012-07-14
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    There will be only two integer cases for $n = 2$ and $n = 3$ and for everything else $$ \sqrt{1\underbrace{4\cdots4}_{n-times}}=2\sqrt{36\underbrace{11\cdots1}_{(n-2)-times}}$$2012-07-14
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    I changed the TeX code from {}_{n-times} to {}_{n\text{ times}, so that instead of $1\underbrace{4\cdots4}_{n-times}$ we see $1\underbrace{4\cdots4}_{n\text{ times}}$. The hyphen looked like a minus sign (longer than a hyphen) since it was in math mode, and the "times" got italicized and needed something to artificially separate it from the $n$, since it was in math mode.2012-07-14

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