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September 28th, 2013, 08:14 AM   #1
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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.
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October 7th, 2013, 02:38 AM   #2
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Re: Rational number

If is rational, must be of the form either
where are rational and is a positive non-square rational, or
where are rational and is a positive non-cube rational. Let us look at each of these cases.

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 non-zero rational solution in so again and is rational.
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