I'm looking at the following equivalence relation on $\mathbb{Z}$: $a \sim b$ if and only if there exist $n,m \in \mathbb{N}_{>0}$ so that $a^n = b^m$
I'm trying to determine what the cardinality of the equivalence classes would be. The equivalence classes of $-1,0,1$ their cardinality are not that hard to determine but when we look at an element $a \in \mathbb{Z}\setminus\{-1, 0, 1\}$ things start to get exciting.
My argument would be that if we look at a random $a \in \mathbb{Z}\setminus\{-1, 0, 1\}$ the cardinality of its equivalence would be at least countably infinite, considering $a \sim a^2 \sim a^4 \sim a^8\cdots$ for all $a \in \mathbb{Z}$.But could it be possible that the equivalence class of such an $a \in \mathbb{Z}\setminus\{-1, 0, 1\}$ would be uncountably infinite?