- #1
Ninty64
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Homework Statement
Show that [itex]x^{n}+y^{n}=z^{n}[/itex] has a nontrivial solution if and only if the equation [itex]\frac{1}{x^{n}}+\frac{1}{y^{n}}=\frac{1}{z^{n}}[/itex] has a nontrivial solution.
Homework Equations
By nontrivial solutions, it is implied that they are integer solutions.
The Attempt at a Solution
I was able to solve in one direction
Given an integer solution to [itex]\frac{1}{x^{n}}+\frac{1}{y^{n}}=\frac{1}{z^{n}}[/itex]
Then it follows that:
[itex]\frac{x^{n}+y^{n}}{x^{n}y^{n}}=\frac{1}{z^{n}}[/itex]
[itex]x^{n}y^{n}=z^{n}x^{n}+z^{n}y^{n}[/itex]
[itex](xy)^{n}=(zx)^{n}+(zy)^{n}[/itex]
Therefore, since the set of integers is closed under multiplication, then [itex]x^{n}+y^{n}=z^{n}[/itex] has a nontrivial solution.
However, I can't seem to prove the other direction. Working the above proof backwards doesn't work unless I assume that d = gcd(x,y) and somehow prove that z = xy.