Under what all conditions on $(a,b,c)$ where $a,b,c$ are positive integers can we say about the non-existence or existence of any solution to the equation $$a^a+b^b=c^c$$
(Or in other words : Solve $a^a+b^b=c^c$ ) $ $
$ $
About this note that : For a solution we must have $gcd(a,b,c) = 1$ (follows from Fermat's Last Theorem)
and all the elements of the set $\{a,b,c\}$ cannot be prime numbers (proof shown below).
We will prove there are no positive prime numbers $p,q,r$ such that the following equation is satisfied : $$p^p+q^q=r^r$$ This is easy to prove. Note that in fact we have something stronger : The equation $4+m^m=n^n$ has no solutions in positive integers. Assume the contrary, then we get $m>1$ (by checking the case $m=1$ separately) and then note that $n \ge m+2$ and thus $n^n > (m+2)^m > m^m+2^m \ge n^n$ and thus a contradiction.