Seems like a problem solved, but I need help.

The point was to prove that $\displaystyle a^n+b^n=c^m$ - with $\displaystyle m,n$ relatively prime - had an infinite number of solution in positive integers $\displaystyle a,b,c$. I plug $\displaystyle a=b=2^x,c=2^y$ to arrive at $\displaystyle nx+1=ym \Leftrightarrow nx-ym=1$. Bezout's lemma tells me this has an infinity of solutions $\displaystyle x,y$ and hence an infinity of solutions $\displaystyle (a,b,c)=(2^x,2^x,2^y)$. But if $\displaystyle m,n$ are of opposite signs, then when $\displaystyle x$ rises, $\displaystyle y$ decreases, and $\displaystyle x,y$ are not always positive $\displaystyle \Rightarrow 2^x,2^y$ are not always integers.

What am I missing?