solution of equation $x^n-y^m=2$

Question

can anyone give me solution of $x^n-y^m=2$ where $x,y$ are positive integers and $m,n$ greater than or equal to $2$ except $(x,y,n,m)=(3,5,3,2)$ ?if this is only solution of given equation,then can anyone prove it ?

Answer 1

This is Pillai's conjecture, e.g., on the integer solutions of $$ x^n-y^m=c $$ with $n,m\ge 2$ and $c\ge 1$. There should be only finitely many solutions, and very often only one - see here, Conjecture $1.6$. There is also an interesting paper by Michael A. Bennett, which shows some more results here. In fact, he proves that $3^3-5^2=2$ is the only solution in positive integers for $c=2$, see Theorem $1.5$.