What does it take to find integer solutions to this exponential division equation?

Question

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.

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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.

Answer 1

There is another representation of this formula, which connects it with an -as well unsolved- "detail in the Waring-problem" (see mathworld.com) namely the distance of $(3/2)^N$ to the next integers.

By rewriting (I use different variables names from your formula to match the notation in an essay of mine:N for b - because "N" indicates the "N"umber of odd steps, "A" for "a" because I use capital letters for exponents in this environment, see section 4.2 in the essay) $$ k(2^A2^N-3^N)=2^A-1 \\ 2^A(2^N k-1)=3^N k-1 \\ $$ $$ 2^A={3^N k-1 \over 2^N k-1} \tag 1$$ Introducing a functional notation $$ f(N,k) = {3^N k-1 \over 2^N k-1} \tag 2 $$ $\qquad \qquad $ relates to a well-known conjecture, but which is again unproven until today.
$\qquad \qquad $ We can reformulate this as $$ f(N,k) = (3/2)^N + (3/4)^N/k - \varepsilon_{N,k} \tag 3 $$ $\qquad \qquad $ where $\varepsilon_{N,k} < 1/2^N$ for $N,k \gt 1$.

Looking at the first part only and omitting the small subtractive summand, we find an expression, which occurs in the "detail" $$ f^*(N,k) = (3/2)^N + (3/4)^N \lt \lceil (3/2)^N \rceil \qquad \text {for } N \gt 7 \tag 4 $$ and that means, that in $(1)$ not only we cannot have a perfect power of $2$ on the lhs, but even not an integer at all, because due to conjecture $(4)$ the next integer above $(3/2)^N$ is larger than $f^*(N,k)$ and thus than $f(N,k)$ as well.

So if the "detail in the Waring-problem" could be solved/proved independently, then again we had the argument against the "1-cycle".

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I didn't investigate the reverse idea: but I think it should be an interesting discussion, whether the Steiner/Simons/deWeger-disproof of the "1-cycle" can be expanded into a formal solution of the "detail in the Waring-problem". Perhaps this is doable with more experience/math-training than I have.

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Appendix: see the picture of $f^*(N,1)$ . (The picture is rescaled for the $\tanh^{-1}()$ of the interval $0 .. 1$ of the resp. fractional parts)

Indeed, for $N \gt 7$ the red points for $f^*(N,1)$ (denoted as $g(N)$ in the picture) lay in the very near of the grey/blue points, and for $N \gt 50$ the different coordinate is practically indiscernable, thus empirically confirming the conjecture $(4)$ from the "detail" up to $N=20000$.

Answer 2

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...

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$\;^{[1]}$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.