quasi
Posts:
10,975
Registered:
7/15/05


Re:     validity of an equation
Posted:
Mar 2, 2013 9:24 AM


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, 2y. > >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

