Question 116
Let $n$ and $m$ be positive integers. We call a set $S$ of positive integers $(n, m)$-good if it satisfies the following three conditions:
(i) We have $m \in S$.
(ii) For all $a \in S$, all of the positive divisors of $a$ are elements of $S$ too.
(iii) For all mutually different numbers $a, b \in S$, we have $a^{n}+b^{n} \in S$.
Determine all pairs $(n, m)$ for which the set of all positive integers is the only $(n, m)$-good set.
(i) We have $m \in S$.
(ii) For all $a \in S$, all of the positive divisors of $a$ are elements of $S$ too.
(iii) For all mutually different numbers $a, b \in S$, we have $a^{n}+b^{n} \in S$.
Determine all pairs $(n, m)$ for which the set of all positive integers is the only $(n, m)$-good set.