Question 23
Let's call a pair of natural numbers $\overline{a_1 a_2 \dots a_k}$ and $\overline{b_1 b_2 \dots b_k}$ $k$-similar if all the digits $a_1, a_2, \dots, a_k, b_1, b_2, \dots, b_k$ are pairwise distinct and there exist distinct natural numbers $m, n$ such that the following equation holds:
$$
a_1^m + a_2^m + \dots + a_k^m = b_1^n + b_2^n + \dots + b_k^n.
$$
What is the largest value of $k$ for which there exist $k$-similar numbers?
$$
a_1^m + a_2^m + \dots + a_k^m = b_1^n + b_2^n + \dots + b_k^n.
$$
What is the largest value of $k$ for which there exist $k$-similar numbers?