A perfect power is a positive integer that can be expressed as an integer power of another positive integer. This tag should only be used when having in mind an arbitrary perfect power (as opposed to a specific one, like a perfect square, a perfect cube, etc.).

Questions about perfect powers, which are defined by:

A perfect power is a positive integer that can be expressed as an integer power of another positive integer.

This tag should only be used when having in mind an arbitrary perfect power (as opposed to a specific one, like a perfect square, perfect cube, etc.).

A related theorem is Catalan's Conjecture (now proved), stating that (given integers $x,y,a,b>1$)

$$x^y-a^b=1\iff (x,y,a,b)=(3,2,2,3)$$

Pillai's Conjecture is a conjecture that concerns whether every difference (not only $1$) of perfect powers occurs only finitely often.