Catalan's Diophantine problem

This article is about an exponential Diophantine equation: a Diophantine equation/problem that involves one or more variables in the exponent.
View a complete list of exponential Diophantine equations

Statement

This problem asks for the solutions to:

$x^p-y^q=1$.

Here, $x,y$ are positive integers not equal to $0,1$ and $p,q$ are positive integers greater than $1$.

Catalan's conjecture was that the only solutions to this are $x=3,y=2,p=2,q=3$. This conjecture has been proved.