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 $\epsilon$, there exists $C_{\epsilon }$ such that if $a+b=c$, then $\max\{|a|,|b|,|c|\}\leq C_{\epsilon }\prod _{p|abc}p^{1+\epsilon }$, the product over $p$ prime