Good day
The problem is as follow:
Find all solutions $(x, y)$, where $x, y \in \mathbb {Z^+}$ to the equation: $$1+3^x=2^y$$
Two solutions are $(0,1)$ and $(1,2)$ but how do you go about calculating other possible solutions or proving those are the only ones? Also, what would the general approach be to solving such an equation? Without a calculator, as you would need to do in an olympiad.
Thank you!
Hint. Examine the possible values modulo $8$ of $ 1 + 3^x$.