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I know this is a special case of a result that works in both directions for the nth root. But I'm trying to find a simple proof of this result.

Assume $gcd(p,q)=1$ and $\sqrt k=\frac{p}{q}$, so $k = \frac{pp}{qq}$ and $qqk=pp$ ---> Then some fundamental theorem of arithmetic magic comes next or something.

Any help would be appreciated

Bill Dubuque
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numbertheory_hobo
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jul 08 '22 at 16:23
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    Most all of the well-known proofs already appear here - see the linked answers (and the "Linked" questions there. – Bill Dubuque Jul 08 '22 at 16:25
  • If $\gcd(p,q)=1$ then they don't have any nontrivial factors in common. What does $k =\frac {p^2}{q^2}$ say about any factors $p$ and $q$ may have in common? – fleablood Jul 08 '22 at 18:51

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