This question arose as an attempt to answer the following question Relaxed Collatz 3x+1 conjecture. I wanted to show that there is a solution of the equation $2^{k}=3^{z}(2n+1)-1$ for each $n\geq 2$, where $k,z,n\in\mathbb{N}$. But even a special case has put me in a dead end.

Are there infinitely many solutions of the equation $2^k=3^z-1$, when $z\rightarrow \infty$? First solution: $2^3 =3^2 -1$.

can occur I don't know at the top of my head, but I think that actually many $w$ don't occur. $\endgroup$ – Gottfried Helms Nov 25 '17 at 11:49every w