Solve equation $ 1+2^x=3^y$ Find integers $x$ and $y$ such that$$ 1+2^x=3^y.$$
It is obvious that $x = y = 1$ and $x = 3, y = 2$ are solutions.
I think others are not. 
How to show that?

Case 1: $y$ is odd.  Then $3^y \equiv 3 \mod 8$.  When is $2^x \equiv 2 \mod 8$?
Case 2: $y = 2 z$ is even.  Then $2^x = 3^{2z} - 1 = (3^z - 1)(3^z + 1)$, so 
$3^z - 1$ and $3^z + 1$ are powers of $2$.  What powers of $2$ differ by $2$?
By the way, this proof of a special case of Catalan's conjecture dates back to Gersonides (Levi ben Gershon) in 1343.
