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So this question was given as a bonus question on my practice exam, but I am interested in solving it...

So if $(x+\frac{1}{x}), x \in \mathbb{R}$ is an integer then show $x^n + \frac{1}{x^n}$ is an integer.

I don't even know where to start, maybe binomial expansion? But that would only give $x^n*\frac{1}{x^n}$, not $x^n + \frac{1}{x^n}$.

Is there something I'm missing (also this is quite a basic course, so as simple as solution as possible please). We have covered basic methods of proofs (including induction). I have read $x+1/x$ an integer implies $x^n+1/x^n$ an integer which of course is the same question, but I don't quite get the how some of the answers got to the induction step.

Thanks

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    $\left(x^n+\frac 1{x^n}\right)\times \left(x+\frac 1{x}\right)=x^{n+1}+\frac 1{x^{n+1}}+x^{n-1}+\frac 1{x^{n-1}}$ – lulu Apr 14 '16 at 12:20
  • In the accepted answer for the linked question, the case for $n4 follows from the cases $n-1$, $n-2$ and $1$. Hence if you have $n=1$ and $n=0$, you are done. – Hagen von Eitzen Apr 14 '16 at 12:21

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