TITLE: Find all positive integers k and l such that $3^k - 2^l = 1$
QUESTION [5 upvotes]: Possible Duplicate:
$|2^x-3^y|=1$ has only three natural pairs as solutions 

As the title says:
Find all positive integers k and l such that $3^k-2^l=1$
In previous questions it is proved that $2^l+1$ is divisible by $27$ iff it is divisible by $19$, though I'm not sure how that is meant to help.
I've brute forced it, and got that $k = l = 1$ works, as does $k = 2$ and $l = 3$, but I can't figure out how to prove that these are the only solutions.
Any help would be  much appreciated

REPLY [2 votes]: If by "iff" you mean "if and only if" (just to be sure, I didn't learn maths with english) then your problem is almost solved :
take your original equation, and pass $2^l$ to the other side of the equal.
$3^k$ and $2^l +1$ are equal if they have the same prime dividers.
What are the prime divider of $3^k$? Only 3.
What are the prime divider of $27$? Only 3.
So you can use your previous questions to find other possibility :
$2^l + 1$ must be a multiple of 19, and must be an odd number.
How many solutions of this sub-question can you find ?
Once you have them it should be easy to find the corresponding value for $k$.