Smallest gaps between powers of 2 and 3 I am trying to find the smallest gaps between powers of 2 and 3. 
Such examples are: (2, 3) (3, 4) (8, 9) (27, 32) (243, 256) (2048, 2187) (16384, 19683) (524288, 531441)... What are the next smallest gaps possible? 
With gaps of 1, 1, 1, 5, 13, 139, 3299, 7133?
I would have checked to see if there is an Oeis sequence for this and there appears to be none. 
Help appreciated. 

By the Mihăilescu-Catalan theorem (but for sure this particular case has an elementary solution)
https://en.wikipedia.org/wiki/Catalan's_conjecture
you'll never get $2^k-3^l=1$ with $k,l>1$ and the only case when  $3^k-2^l=1$ for $k,l>1$ is $3^2-2^3$.
Hence $3-2$, $4-3$, $9-8$ are the only cases when you get 1 as a difference.

Big differences seem to cause a highly nontrivial problem. See
https://terrytao.wordpress.com/2011/08/21/hilberts-seventh-problem-and-powers-of-2-and-3/
https://mathoverflow.net/questions/116840/distance-between-powers-of-2-and-powers-of-3
