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**tony123****Member**- Registered: 2007-08-03
- Posts: 189

Determine all real x

such that

is an integer for any integer n.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Sorry

I thought I got *x* = ±1, but my working was wrong.

*Last edited by JaneFairfax (2008-03-19 09:20:34)*

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

assume that x+1/x is an integer. Squaring gives:

which also is an integer

now assume that

are both integers

the following product is an integer:

therefor, x^(n+1) + x^(-n-1) is an integer and by the induction principle it holds for all n.

Hence all x that satisifies the condition is the ones that satisfy that x+1/x is an integer, which implies that there exists infinite many reals satisfying the problems condition.

ps. how do i do the sign, that shows that an element belongs to a set in mathcode?

*Last edited by Kurre (2008-03-23 01:45:43)*

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

Thanks!

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