Does $a^a+b^b=c^c$ have solutions over positive integers?
I did not try almost anything because I do not know how to handle these kind of questions.
I did not find this question anywhere, just created it so that I have something to calculate and think about, but, unfortunately, I do not know enough of number theory to attack this problem.
Do you have any ideas?
It could be that this is not at all hard, but at this moment I do not have a clever idea.
There are no solutions over the positive integers.
Without loss of generality, $a \le b$. Then the left hand side is bounded by $2b^b$; on the other hand, $c > b$ and the right hand side is therefore at least
$$c^c \ge (b + 1)^{b + 1} > b^b \cdot (b + 1) \ge 2 b^b$$
giving a contradiction.