\[n^5+n-1=(n^2-n+1)(n^3+n^2-1)\]
(This could be manually done through comparing coefficients.)

The next thing to note is that they have to be both perfect powers to avoid having two distinct factors. We can easily check from here that they can't be perfect powers and thus they must have atleast two distinct factors.

I can post the proof on request but for now it is left as an excercise to the reader.

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TopNewestHint: The polynomial can be factored.See answer here.

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The factoring is really easy

\[n^5+n-1=(n^2-n+1)(n^3+n^2-1)\] (This could be manually done through comparing coefficients.)

The next thing to note is that they have to be both perfect powers to avoid having two distinct factors. We can easily check from here that they can't be perfect powers and thus they must have atleast two distinct factors.

I can post the proof on request but for now it is left as an excercise to the reader.

@Sharky Kesa now onto your functional equation.

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If you want, you can post the final part of the proof as a DM to me on Slack.

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