If $ab > 1$, can $\frac{3^a - 2^a}{2^{a+b} - 3^a}$ be an integer? I've been thinking about this positive fraction where $a,b$ are positive integers and whether it can be an integer for $ab > 1$:
$$\frac{3^a - 2^a}{2^{a+b} - 3^a}$$
Clearly, if $a=1, b=1$, then:
$$\frac{3^1-2^1}{2^2 - 3^1} = 1$$
Is there any other value of $a,b$ that results in a positive integer?
I am interested in either finding a solution where $ab > 1$ and it is a positive integer or an approach to show that no such integer exists.
 A: You've reminded me of my least favorite fact about the fundamental theorem of arithmetic, which is that it ruins music.
We need the denominator to be (relatively) small if it's going to be a divisor of anything. But small values in the denominator are going to be hard, if not impossible, to find, as $a$ increases. Obviously we cannot have $2^x = 3^y$ unless $x = y$. We have $3^2 - 2^3 = 1$ from Catalan's Conjecture, but that's in the wrong order.
But then it gets worse. The generalized/extended Catalan Conjecture suggests that prime powers have a tendency to repel one another as they get larger. For instance, there are exactly five instances below $2^64$ where a pair of prime powers fall between one prime and the next. The pairs of powers are $(8,9)$ as above, $(25,27),(121,125),(2187,2197)$, and $(32761,32768)$.
The smallest positive value of $2^{a+b} - 3^a$ for a given $a$ (as found by varying $b$) isn't technically an increasing function, but the decreases are few and far between. And decreasing from $24062143$ to $5077565$ (for $a = 16,17$) isn't what we're looking for. The only "small" values other than your $a = b = 1$ are $5,7,13,47$ for $a = 3,2,5,4$. None of those give an integer quotient (they are $\frac{19}{5}, \frac{65}{7}, \frac{211}{13}, \frac{665}{47}$), and after that most of the values just get exponentially larger.
Essentially, barring an exception to the extended Catalan Conjecture, which I think would be hugely shocking, you can't get an integer out of this. But, technically, it's an open question.
