For which integers $m$ is there a solution to $3^y m - 2^x = 1$? This self-answered question is motivated by a recent question asking whether for every $m$ there is a solution $(x, y)$ of integers to the equation $$3^y m - 2^x = 1$$ for every odd positive integer $m$. As the answers there show there is not---the easiest way to see this is to recall that the only solutions to $3^{y'} - 2 x = 1$ have $y' \leq 2$, so for $m = 3^k$, $k > 2$, the resulting equation $3^{y + k} - 2^x = 1$ has no solutions. But this raises the natural question:

For which integers $m$ is there a solution $(x, y)$ in positive integers to the equation $$3^y m - 2^x = 1 \,?$$


Reducing modulo $3^y$ and rearranging leaves $$2^x = -1 \pmod {3^y} .$$ On the other hand, since $2$ is a primitive root modulo $3^1, 3^2$, it is a primitive root modulo $3^y$ for all $y \geq 1$. So, since $(\Bbb Z / (3^y \Bbb Z))^\times$ has order $\phi(3^y) = 2 \cdot 3^{y - 1}$, for fixed $y$ the $x$ satisfying the above congruence are exactly those of the form $$x = \frac{\phi(3^y)}{2} (2 k + 1) = 3^{y - 1} (2 k + 1) , \qquad k \in \Bbb Z_{\geq 0} .$$
Thus, substituting in the original equation and solving for $m$ gives that the $m$ that occur are exactly those of the form
$$\color{#df0000}{\boxed{m = \frac{2^{3^{y - 1} (2 k + 1)} + 1}{3^y}, \qquad k \geq 0, y \geq 1}} .$$
The solutions with $\color{#df0000}{m} < 1000$ give the equations
\begin{alignat}{4}
3^1 \cdot \color{#df0000}{1} - 2^1 &= 1 &
3^2 \cdot \color{#df0000}{1} - 2^3 &= 1 & 
3^1 \cdot \color{#df0000}{3} - 2^{3\phantom{1}} &= 1 \\
3^1 \cdot \color{#df0000}{11} - 2^5 &= 1 &
3^3 \cdot \color{#df0000}{19} - 2^9 &= 1 &
3^1 \cdot \color{#df0000}{43} - 2^{7\phantom{1}} &= 1 \\
3^2 \cdot \color{#df0000}{57} - 2^9 &= 1 &\qquad
3^1 \cdot \color{#df0000}{171} - 2^9 &= 1 &\qquad
3^1 \cdot \color{#df0000}{683} - 2^{11} &= 1 .
\end{alignat}
The smallest $\color{#df0000}{m}$ giving a solution of the form $(x, 4)$ yields
$$3^4 \cdot \color{#df0000}{1\,657\,009} - 2^{27} = 1 .$$
Incidentally, the sequence $1, 3, 11, 19, 43, \ldots$ of $\color{#df0000}{m}$ that admit solutions does not appear in the OEIS.
