Determine all integers $x $ and $ y$ such that $|2^x − 3^y| =1$ I am having trouble solving this problem:

Determine all integers x and y such that  $|2^x − 3^y| =1$

I would think that the only solutions to it is $x = y = 1$.
How can I show that there is no other solutions?
If there are other solutions, how can I find all the solutions?

Hint: If $2^x - 3^y = 1$, then $2^x = 1 + 3^y$. Take both sides modulo $16$:
$$2^x \equiv 1 + 3^y \pmod{16}.$$
If $x\ge 4$, then the left-hand side is $0$. The sequence of $3^y \pmod{16}$ is $3,9,27=11,81=1,3,9,11,\ldots$, so the right-hand side is $1+3=4$, $1+9=10$, $1+11=12$, or $1+1=2$, none of which is equivalent to $0\pmod{16}$. Hence, there are no solutions when $x\ge 4$.
Now, what happens if $2^x - 3^y = -1$?
