Powers Which Differ by a Value of One

[CarsonAdams]

Are there natural solutions for A and B which satisfy the equation

A^x+1=B^y

where x and y are integers greater than 2?
(I only include a greater than 2 stipulation because I can see a few obvious solutions such as 2³+1=3²)

This has just sort of come to mind as I was thinking about the interesting differences in the prime composition of numbers differing by one, and then wondered if there were any powerful primes that differed by only one, or any powerful numbers whatsoever that differ by 1.

 

[mfb]

The non-existence of such numbers is Catalan's conjecture, and its proof is quite new.