Catalan's Conjecture

Theorem
The only solution to the Diophantine equation:
 * $x^a - y^b = 1$

for $a, b > 1$ and $x, y > 0$, is:
 * $x = 3, a = 2, y = 2, b = 3$

Also known as
This result is also known as Mihăilescu's Theorem, for.

Also see

 * Consecutive Integers which are Powers of 2 or 3: the special case where $x$ and $y$ are $2$ and $3$
 * 1 plus Power of 2 is not Perfect Power except 9: the special case of $y = 2$.