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September 28th, 2013, 08:14 AM  #1 
Member Joined: Jun 2013 Posts: 31 Thanks: 0  Rational number
Let x be a real number with the property: x^3+x and x^5+x are rational. Prove, that x is rational.

October 7th, 2013, 02:38 AM  #2 
Member Joined: Mar 2013 Posts: 90 Thanks: 0  Re: Rational number
If is rational, must be of the form either Case 1: We have For this to be rational, the coefficient of must vanish. Hence (since ) and so is rational. Case 2: We have . Then yields or . (i) This leads to and . If are not both zero, this means must take one of the following forms: if if Take the first one. Now examine the coefficient of in . It works out as . Then the coefficient of in is . Since this cannot be zero, is irrational. But if and are both rational then their difference must be rational as well. This contradiction shows that is impossible. The other one leads to a similar impasse. Thus if , we must have and so which is rational. (ii) Since is not a perfect cubic rational, this has no nonzero rational solution in so again and is rational. 

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