I was looking for a natural power of 3 that could be written like 

> Binary format:
> 
> 11..(N times)..11011..(M times)..11
>
> Example: 1111110111111111111111 (...isn't a power of 3)

Or could also be written like 

> 3^x = 2^a - 2^b - 1
> 
> (x is arbitrary, "a" and "b" are natural numbers, a = N-M-1, b = M, a-b >= 2)

But couldn't find any, so I thought there might be some proof that there's no such numbers (altho that would contradict intuition) or maybe it can be proven that there might be such numbers?