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Topic: ---- --- --- --- validity of an equation
Replies: 5   Last Post: Mar 14, 2013 6:20 PM

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quasi

Posts: 10,325
Registered: 7/15/05
Re: ---- --- --- --- validity of an equation
Posted: Mar 2, 2013 9:24 AM
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Deep wrote:

>Consider the following equation under the given conditions
>
>y^(1/2) = (x^10 - z^ 5)^(1/5 ) (1)
>
>Conditions: y, x, z are co prime integers each > 5, 2|y.
>
>Question: can (1) have any solution ?


There are no solutions.

Suppose otherwise.

Raising both sides to the 10'th power yields


y^5 = (x^10 - z^5)^2

Letting w = x^10 - z^5, we get

y^5 = w^2

Since w^2 is a perfect 5'th power, and since gcd(2,5) = 1,
it follows that w is a perfect 5'th power. Thus, we can
write w = t^5 for some integer t.

The hypothesis implies that w is nonzero, hence t is also
nonzero. Then

w = x^10 - z^5

=> t^5 = x^10 - z^5

contrary to Fermat's Last Theorem for exponent 5.

quasi



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