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# Proof Problem

Let $$x$$ be a real number such that $$x + x^{-1}$$ is an integer. Prove that $$x ^n + x^{-n}$$ is an integer, for all positive integer $$n$$.

Note by Dev Sharma
1 year, 1 month ago

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It's probably easiest to use strong induction. The statement is given for $$n = 1.$$ Now suppose $$x^{k} + x^{-k}$$ is an integer for all $$k \le n.$$ We have that

$$\left(x^{n} + \dfrac{1}{x^{n}}\right) \left(x + \dfrac{1}{x}\right) = x^{n+1} + \dfrac{1}{x^{n-1}} + x^{n-1} + \dfrac{1}{x^{n+1}}$$

$$\Longrightarrow x^{n+1} + \dfrac{1}{x^{n+1}} = \left(x^{n} + \dfrac{1}{x^{n}}\right)\left(x + \dfrac{1}{x}\right) - \left(x^{n-1} + \dfrac{1}{x^{n-1}}\right),$$

which is an integer by the induction assumption. This completes the proof by strong induction. · 1 year, 1 month ago

Nice one sir :)

Thanks for the proof :) · 1 year, 1 month ago