Are there any solutions to $\frac{3^n - 2^n}{2^k-3^n} = N$ for $n$, $k$, $N$ $\in\mathbb{N}$, greater than 2.

This is related to a previous answered question: Are there any solutions to $2^n-3^m=1$

Things I already know:

There are no appropriate solutions to $2^k-3^n = 1$ which would make this trivial.

There is an equivalent question which is $\frac{2^{k-n} - 1}{2^k-3^n} = M$.

Things I've tried:

Several variants on diophantine equations. I've not yet found anything that involves quotients, etc. any pointers here would be welcome.

Several things to do with modulus/cyclic groups. The best outcome of this has been the above alternate form which doesn't seem any closer really.

Experimentally, there are no solutions with $n, k < 1000$. There are some pretty tight limits on what k can be as you need $2^k-3^n < 3^n - 2^n$ but also $2^k > 3^n$ which is only one or two $k$'s for every $n$. Also so far I've not seen any $N > 16$ which is a bit puzzling given the size of the values in the fraction. This makes me think there may be a limit based argument. Any pointers to where to start here would also be welcome.


It is not too hard to show that there are no solutions. The assumption that $2^k-3^n$ divides $3^n-2^n$ implies that the former quantity divides $2^{k-n}-1$ and, further, that $2^k < 3^{n+1}$, so that $k < (n+1) \log (3)/\log(2)$. We thus have $$ \left| 2^k - 3^n \right| < 2^{(n+1) \log (3)/\log(2)-n}. $$ On the other hand, standard lower bounds for linear forms in $2$ logarithms show that $$ \left| 2^k - 3^n \right| \geq \min \left\{ 2^k, 3^n \right\}^{0.9}, $$ say, with precisely $23$ exceptions (this is a result from de Weger's thesis; the largest exception is with $(k,n)=(84,53)$). Putting these inequalities together gives you what you want.


Start with Richard Guy's Unsolved Problems In Number Theory. One problem which is considered is how small can the difference be between 2^k and 3^n with respect to the smaller of the two numbers. I suspect Anand Pillai had some results in this direction, but Guy should have some good references to follow. The divisibility problem may have also been considered; use a citation index once you have a good starting point.

Gerhard "Email Me About System Design" Paseman, 2011.07.07


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