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 $n$ .

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

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.

Brian Charlesworth - 5 years, 8 months ago

nice

Dev Sharma - 5 years, 8 months ago

Nice one sir :)

Thanks for the proof :)

Mehul Arora - 5 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$