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Let $a,b$ be positive integers satisfying $(ab)^{n-1}+1 \mid a^n +b^n.$ Then how to show that the number $\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power of an integer?

Another question: is this problem true for $a,b \in \mathbb Z$?

PS. posted to be connected with this.

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    Yes, I know about the AoPS page, but I liked to see other solutions. What about that "Another question"?2011-04-09

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