Does equation $a^b+b^c+c^a=d^e$ has solution s in $\mathbb {N}$
My trial: If there is no restriction for equality of a, b, c , d and e then for these numbers all $\leq 10$ we have 36 set of solutions such as (2, 2, 4, 6,2), (1, 2, 1, 2, 2).I found these by a simple Python program.I have two questions:
1- How one calculates the number of sets of solutions for all numbers in equation $\leq n$, for example $n=10$?
2- Has this equation solutions in $\mathbb{N}$ when $a\neq b$,$b\neq c$, $c\neq d$, $d\neq e$, $e\neq a$ when $a, b, c, d, e> 1$?