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Suppose we have sets on both equalities.....

$$\left\{\frac{i^{\gcd(a,b)}}{j^{\gcd(a,b)}}:i,j\in\mathbb{Z}\right\}=\left\{\frac{p^{a}}{q^{b}}:p,q\in\mathbb{Z}\right\}$$

For fixed $a,b\in\mathbb{N}$

What would be the values of $p$ and $q$ when $i$ and $j$ Is fixed?

It seems the answer to this post would not work.

For example if $a=1$, $b=5$, $i=2$, $j=3$, the answer to this post states that $x+5y=1$ so “I assume” $x$ can equal $-4$ while $y$ can equal $1$ hence $p=2^{1} 3^{4}=162$ and $q^5=(2^{4} 3^{1})^5$ since $p/q \neq {2/3}$ we can conclude the proof does not help us solve our problem.

What can we use instead? Any hints?

Arbuja
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    There's a typo in that answer, which I just noted on the linked question. There seems to be a more basic problem, which I also noted. – saulspatz Sep 23 '19 at 15:28
  • Actually, I think the answer given there had two typos, which offset when both are corrected. Make the corrections I suggested there, and I think things will work. – saulspatz Sep 23 '19 at 15:35
  • This question is best answered in the other post. We can essentially close this as a duplicate or obsolete. – Erick Wong Sep 24 '19 at 14:44

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