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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, 8 months 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. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth Nice one sir :)

Thanks for the proof :) Mehul Arora · 1 year, 8 months ago

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@Brian Charlesworth nice Dev Sharma · 1 year, 8 months ago

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