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Topic: Decomposition of a 10th degree equation
Replies: 4   Last Post: Mar 16, 2013 10:17 AM

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Pubkeybreaker

Posts: 1,392
Registered: 2/12/07
Re: Decomposition of a 10th degree equation
Posted: Mar 15, 2013 9:09 AM
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On Mar 15, 7:32 am, Deep <deepk...@yahoo.com> wrote:
> Consider the following equation (1) for the given conditions.
>
> x^10 + y^10 = z^10                  (1)
>
> Conditions: x, z are odd integers > 0 and y is non integer but x^10, y^10, z^10 are all integers each > 0
>
> (1) can be decomposed as (2) and (3) where x = uv and u, v are co prime integers.
>
> z^5 + y^5 = u^10           (2)                z^5 - y^5 = v^10          (3)
>
> It is seen that if (2) and (3) are multiplied (1) is obtained.
>
> Question: Is the  decomposition of (1) into (2) and (3) valid?
>
> If not why not.
> Any helpful comment will be appreciated.


(A) If (2) and (3) are true, then (1) is true.

This is correct.

But the converse need not be. The converse says:

(B) If (1) is true, then (2) and (3) are true.

And this is an invalid conclusion.



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