A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to whether or not there are integer solutions to the equation.
More importantly though, the mention of Fermat's last theorem got me thinking. Might there be an analogue of Fermat's last theorem for tetration?
Specifically, it occurs to me the equation $a^a+b^b=c^c$ is the case of ${^n}a+{^n}b={^n}c$ for which $n=2$. If there are integer solutions for $n=2$, then what about $n=3$? $n=4$? $n=1397$?
I would guess that there are only integer solutions if $n=1$, but is this the case? And what about subsequent hyperoperations?
WLOG assume $a\leq b$ then $c\geq b+1$ and $^nc\geq {^n} (b+1)>{^n} b+{^n} b\geq{^n} b+{^n} a$.