MathDB

Let's call a pair of positive integers

\overline{a_1a_2\ldots a_k}

and

\overline{b_1b_2\ldots b_k}

k

-similar if all digits

a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k

are distinct, and there exist distinct positive integers

m, n

, for which the following equality holds:

a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n

For which largest

k

do there exist

k

-similar numbers?
Proposed by Oleksiy Masalitin

Problem Statement