Re: Rational and irrational numbers

On 28 Aug 2005 11:07:03 -0700, deepkdeb@xxxxxxxxx wrote:

>Correction and modification:
>
>Given situations:
>
>sqrt(a) + sqrt(b) = u^k (1); sqrt(a) - sqrt(b) = v^k (2)
>uv is an integer, odd k > 5, both a and b are nonsquare integers > 0.
>
>Assertion: u and v must be of the form (3) and (4) where
>u = A*sqrt(g) + B*sqrt(h) (3); v = A*sqrt(g) - B*sqrt(h) (4)
>g, h are nonsquare integers and A, B are integers > 0.

You are going to need some more corrections ...

Let a=b=2, k=7, v=0, u=8^(1/14).

Then your hypothesis is satisfied but your conclusion is false.

To check the falsity of the conclusion, note that if u has the form

u = A*sqrt(g) + B*sqrt(h)
where g, h are nonsquare integers and A, B are integers > 0

then u is at most degree 4 over Q (either degree 2 or degree 4 to be
precise).

On the other hand, u=8^(1/14) satisfies the irreducible polynomial
x^14-8 over Q, so u has degree 14 over Q.

I don't think you'll be able to repair the hypothesis so easily -- my
intuition is that the conclusion is much too strong for the given
info.

quasi
.

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