Are there any positive integer solutions to $2^n-3^m=1$ with $n,m>2$ ?

By way of justifying the question, I've found lots of info on what happens where $m=n$ (mostly FLT variantions, Darmon + Merel,...), but I don't really know where to look for $m\not=n$.

Also it's pretty obvious that you can't have solutions to similar equations, e.g., $2^n-3^m=2$. There are no solutions for $n,m<1000$ aside from $n=2, m=3$. It seems pretty likely to me that it should happen for some large numbers at some point though.

Are there any theorems I don't know about regarding primes $p,q$ and $p^n-q^m=k$, $k \in \mathbb{N}$ that might rule out a solution or help me find one?

Are there any positive integer solutions to $2^n-3^m=1$ with $n,m>2$ ?

By way of justifying the question, I've found lots of info on what happens when $m=n$ (mostly FLT variations, Darmon + Merel,...), but I don't really know where to look for $m\not=n$.

Also it's pretty obvious that you can't have solutions to similar equations, e.g., $2^n-3^m=2$. There are no solutions for $n,m<1000$ aside from $n=2, m=3$. It seems pretty likely to me that it should happen for some large numbers at some point though.

Are there any theorems I don't know about regarding primes $p,q$ and $p^n-q^m=k$, $k \in \mathbb{N}$ that might rule out a solution or help me find one?

Are there any positive integer solutions to $2^n-3^m=1$ with $n,m>2$ ?

By way of justifying the question, I've found lots of info on what happens where $m=n$ (mostly FLT variantions, Darmon + Merel,...) but don't really know where to look for $m\not=n$.

Also it's pretty obvious that you can't have solutions to similar equations, e.g. $2^n-3^m=2$. There are no solutions for $n,m<1000$ asside from $n=2,m=3$. It seems pretty likely to me that it should happen for some large numbers some point though.

Are there any theorms I don't know about regarding primes $p,q$ and $p^n-q^m=k$, $k \in \mathbb{N}$ that might rule out a solution or help me find one?

Are there any positive integer solutions to $2^n-3^m=1$ with $n,m>2$ ?

By way of justifying the question, I've found lots of info on what happens where $m=n$ (mostly FLT variantions, Darmon + Merel,...), but I don't really know where to look for $m\not=n$.

Also it's pretty obvious that you can't have solutions to similar equations, e.g., $2^n-3^m=2$. There are no solutions for $n,m<1000$ aside from $n=2, m=3$. It seems pretty likely to me that it should happen for some large numbers at some point though.

Are there any theorems I don't know about regarding primes $p,q$ and $p^n-q^m=k$, $k \in \mathbb{N}$ that might rule out a solution or help me find one?