Re: Rational and irrational numbers


quasi wrote:
> 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.
With the additional conditions (a,b) =1 and (g,h) = 1 and abgh > 0 you
will discover that assertion is correct. You will not be able to find
any counter example.

>
> quasi

.

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