Solving $2^{x-z} + 2^{y-z} = 1$ Suppose we have an equation $2^{x-z} + 2^{y-z}  = 1$,where $x, y, z$ are integers from $0$  to $9$.  $x, y, z$ can have same values. I guess $x$ and $y$ should have $9$ possible values like $x=y=0$ and $z=1$ and so on . Or should it have more values? 

Another attempt:
$2^x+2^y=2^z;$
$2^z-2^y=2^x;$
Note: $z>y$;
$z=y+a$, $a$, positive integer.
$2^y(2^a-1)=2^x;$
Implies $a=1$(Why?).
Hence $x=y$; and  $z=y+1$.
The possible cases are?
Check: $2^x+2^x=2^{x+1}=2(2^x)$.
