Catalan's conjecture

Statement

This conjecture states that the solution set to Catalan's Diophantine problem:

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

for $x,y$ positive integers not equal to $0,1$ and $p,q$ positive integers greater than one, has precisely one solution: $x=3,y=2,p=2,q=3$.

The conjecture has been proved.

Relation with other facts/conjectures

Conjecture	Statement of conjecture	Status
Fermat-Catalan conjecture	$a^m+b^n=c^k$ has only finitely many solutions for $a,b,c$ positive integers and $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}<1$	open
abc conjecture	For every $ϵ$, there exists $C_{ϵ}$ such that if $a+b=c$, then $max\{|a|,|b|,|c|\}\le C_{ϵ}{\prod }_{p|abc}{p}^{1+ϵ}$, the product over $p$ prime