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?



Prove that the equation $2^a - 2^b - 1=3^c$ has no integral solution with $a,b\geq 3$
The equation $2^a-2^b-1=3^c$ has no integral solution with $a,b\geq 3$. Indeed, in this case the left-hand side is $\equiv 7\pmod{8}$, while the right-hand side is either $\equiv 1\pmod{8}$ or $\equiv 3\pmod{8}$.


