TITLE: Find $x$ and $y$ in $2^{x-y} + 1 = 2^x,$ where $x,y$ are integers
QUESTION [8 upvotes]: I have no idea what to do now. 
Is there any way to find the integers $x$ and $y$ by factoring?
Thank you.

REPLY [20 votes]: Dividing by $2^x$ gives
$$
2^{-x}+2^{-y}=1
$$
Since $2^{x}\gt0$ for any $x$, we must have both $2^{-x}\lt1$ and $2^{-y}\lt1$; that is, $x\gt0$ and $y\gt0$. Since $x$ and $y$ are integers, the biggest that $2^{-x}$ or $2^{-y}$ can be is $\frac12$ and since their sum is $1$, they must both be $\frac12$.  That means $x=y=1$.