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#### lfdahl

##### Well-known member

- Nov 26, 2013

- 719

$$x^n+x^{-n}$$

is an integer for any integer $n$

Source: Nordic Math. Contest

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- #1

- Nov 26, 2013

- 719

$$x^n+x^{-n}$$

is an integer for any integer $n$

Source: Nordic Math. Contest

- Aug 6, 2015

- 271

- Mar 31, 2013

- 1,283

zero is not valid. so x = -1 or 1

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- #4

- Nov 26, 2013

- 719

Hi, Monoxdifly

Your conclusion is wrong. Try e.g. the value: $x = 2 + \sqrt{3}$. (and $x=0$ is not valid as pointed out by kaliprasad )

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- Feb 7, 2012

- 2,684

$$x^n+x^{-n}$$

is an integer for any integer $n$

Source: Nordic Math. Contest

So it is sufficient to find all $x$ such that $x+x^{-1} = k$. But that is a quadratic equation with solutions $x = \frac12\bigl(k \pm\sqrt{k^2-4}\bigr)$. Those solutions are real for all integers $k$ apart from $k = 0$ or $\pm1$.

Thus the general solution is $x = \frac12\bigl(k \pm\sqrt{k^2-4}\bigr)$ ($k\in\Bbb{Z},\ |k|\geqslant2$).

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