Solving the exponential Diophantine equation $2^{m+1} = zk + 1$ where $m,k,z \in Z$ for a given very large $z$

Question

I am working on a problem where I have ended with an exponential Diophantine equation of the form

$$2^{m+1} = zk + 1$$

where $m,k,z \in Z$ for a given very large $z$ (i.e., factoring $z$ is practically hard).

Are there any techniques I could use to solve this?

Answer 1

Comment:

I think all Mersenne numbers ($2^k-1$) which are not primes can give a family of solutions in $mathbb N$. For example Mersenne number $M_{11}=2047=23\times 89$ which gives:

$(n. z, k)=(10, 23, 89)$