What values can $2^j-3^k$ have?

Question

What values can $2^j-3^k$ have?

E.g., $$ 2^2-3^1=1\\ 2^2-3^0=3\\ 2^3-3^1=5\\ 2^4-3^2=7 $$

Can every number not divisible by $2$ or $3$ be written as $2^j-3^k$? If not, why?

Answer 1

we try to find an $n$ coprime to $6$ such that $2^j-3^k$ doesn't cover all options $\bmod n$.

Since we want $2^j$ to cover a small number of cases we are going to try with $n=2^m-1$.

We find that the order of $3\bmod 511$ is $12$ and the order of $2\bmod 511$ is clearly $9$.

Thus only a small fraction of residues $\bmod 511$ are covered by $2^j-3^k$.