Special case of Pillai's conjecture

Question

Pillai's conjecture is a generalization of Catalan's conjecture. It's say that for fixed positive integers $A, B, C$ the equation $Ax^n - By^m = C$ has only finitely many solutions $(x,y,m,n)$ with $(m,n) ≠ (2,2)$.
But if we fix $A=B=1$ and $x=2,y=3$ then we have :
$$2^n-3^m=C$$

My question : Have this equation only finitely many solution for any fixed integer $C$?

For $C=1,-1$ you can see this answer.

Answer 1

Yes, this follows from the following more general result by taking $P = \{2,3\}$:

Let $P$ be a finite set of primes, and let $S$ be the set of natural numbers whose prime factors are entirely contained in $P$. Then the gaps between elements of $S$ grows to infinity, that is for each $k$ there are only finitely many distinct pairs of $x,y \in S$ such that $|x-y| < k$.

In other words, the set of $r$-smooth numbers for any fixed $r$ is very sparse. This result can be deduced easily from standard theorems on finiteness for high-genus polynomial Diophantine equations. It was previously discussed on this site here:

Gap between smooth integers tends to infinity (Stoermer-type result)?

And some of the details were described in my answer to a related question:

https://math.stackexchange.com/a/725149/30402

Answer 2

Not an answer, but a start.

You can show that, if there is such a $C$ with infinitely many solutions, then there is a constant $K$ such that for all $m,n$ in your solution set:

$$\left|\frac m n-\log_2 3\right|<\frac{K}{n3^n}$$

This means that the continued fraction expansion for $\log_2 3$ must have lots of absolutely huge coefficients.

I think you might be able to prove the converse - if there is such a $K$ and infinitely many $m,n$ satisfying this inequality, then there is a $C$ with infinitely many solutions.

Answer 3

The following paper, Pillai's conjecture revisited by Michael A. Bennet answers this question as yes, there is at most one solution for any c in your question apart from c=-1,5,13 which each have 2 solutions. https://reader.elsevier.com/reader/sd/pii/S0022314X02000495?token=5A3168779EC465B20F7BDA91EA5E568C90D1F648FCF17363B85AD8B8698813DC689A589AECE08EEE82C09486EAD4FAB6&originRegion=eu-west-1&originCreation=20220121224632