TITLE: What does it take to find integer solutions to this exponential division equation?
QUESTION [6 upvotes]: Consider this equation where $a$ and $b$ are positive integers.
$$k = \frac{2^a - 1}{2^{a+b} - 3^b}$$
This equation has the trivial solution $k=1, a=1, b=1$.
How would I find more solutions, or show that no more exist? I'm not asking anyone to solve it, but just explain how I might explore the solutions myself.

I came up with this equation as one whose solution would imply one specific kind of cycle in the Collatz iteration. Since the Collatz conjecture is believed to hold, I expect that there is proof that this equation has no other solutions, and am interested to see what mathematical techniques can be used to eliminate just this one case.

REPLY [2 votes]: Looking at old entries in my literature-database, I found an interesting limiting formula for a lower bound of $2^{a+b}-3^b$. Some short tinkering with it seem to show, that you can prove your conjecture for all $a+b > 27$ with it.
The formula is (Stroeker/Tijdeman,'71): $$ \mid 2^x - 3^y \mid \gt \exp(x (\log 2- \frac1{10})) \qquad \text{for all } x,y \in \mathbb N  \quad \text{and } x\gt27 \quad \;^{[1]}\tag 1$$
This can be applied to your equation. By (1) we can write
$$ 2^{a+b}-3^b \gt \mu ^{a+b} \qquad \text{where } \mu =1.80967483607...  \tag 2$$
and thus
$$ k = { 2^a-1\over 2^{a+b}-3^b} \lt { 2^a\over \mu^{a+b}} \qquad \text{for }a+b\gt 27 \tag 3$$
Here the rhs can be found to be smaller than $1$ for $(a+b) \gt 27$:
$$ \text{(rhs)}=\exp( a\cdot \ln2 - (a+b)(\ln2 - 1/10))\\
=\exp(0.1 a-( \ln2-0.1)b) \\
\approx \frac{1.1^a}{1.8^b} \lt 1 \tag 4$$
and thus we can conclude
$$\implies k \lt 1\qquad \text{for }a+b\gt 27 \tag 5$$
$\qquad\qquad\quad$(Hope I didn't mess with signs and computation...)
Someone might even brush this up a bit...

$\;^{[1]}$The citation of formula (1) is from
R.J.STROEKER & R.TIJDEMAN 
Diophantine equations (with appendix by P.L.Cijsouw, A.Korlaar & R.Tijdeman) 
in: MATHEMATICAL CENTRE TRACTS 154, COMPUTATIONAL METHODS IN NUMBER THEORY; PART I;
MATHEMATISCH CENTRUM, AMSTERDAM  1982

and they attribute this result to W.J.Ellison in 1970/1971
[25] ELLISON,W.J., Recipes for solving diophantine problems by Baker's method,
Sèm.Th.Nombr.,1970-1971,Exp.No.11, Lab.Thèorie Nombres,
C.N.R.S.,Talence,1971.