<s>[INST] 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? 
 [/INST] The only possible case is $x-z = -1 = y-z\implies x = y = z-1$. Thus $z = 1,2,3,4,5,6,7,8,9$, and $x = y = 0,1,2,3,4,5,6,7,8$
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