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, and the single zero in binary format is a must)

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?