(Inspired by the Reddit post here)
Let $m, n, k$ be integers with $m, n>2$ and $k\ge 2$. I'm wondering what we can say about solutions of the Diophantine equation $a^m+b^n = c^k$. Of course, when $\gcd(m, n, k)>2$ this has no solutions by Fermat's Last Theorem.
Assuming $abc$ conjecture, we know for any $\epsilon$ that $c^2<_\epsilon \mathrm{rad}(abc)^{1+\epsilon}$ for all but finitely many coprime $(a, b, c)$. It seems to me that we can choose some $\epsilon$ so that the RHS is $<c^2$ and hence for all but finitely many coprime $(a, b, c)$, though I'm not sure how to get there. Is my intuition correct? If so, how can we approach this (and is there a simpler approach)?