I am looking for a source of the list of known solutions of:
$$2^x-3^y=z$$
with $x, y, z$ integer, $x, y, z > 0$ and $z$ "small". I would like to know especially if there are non-trivial (by trivial I mean small $x$ and $y$) known solutions such that $z < 2^{x-y}$, or if not, if it can be proven that there are no (non-trivial) solution of that form.
Thank you!
Write it as $2^x-2^{y\log_2 3}=z$ and you can see you want $y \log_2 3$ to be close to a whole number. That suggests looking at the convergents of the continued fraction to $\log_2 3$. Of course there is $2^3-3^2=-1$, so if you take the absolute value of $z$ you are right on the border. The next one is $2^8-3^5=256-243=13$ which you wanted less than $2^3=8$. Looking down the line there is a large value of $23$ that would give us $2^{1054}-3^{665}$ if we stopped just before. The difference is about $2^{1039}$, much too large. I don't know how to prove there aren't any, but it looks unlikely. Since $y\approx \frac x{\log_2 3}$ you are asking that $2^x-3^y \lt 2^{x(1-\frac 1{\log_23})}\approx 2^{0.369x}$, which is very small compared to $2^x$