IR-TST 2022 Problem 10

We call an infinite set $S\subseteq\mathbb{N}$ good if for all parwise different integers $a,b,c\in S$ , all positive divisors of $\frac{a^c-b^c}{a-b}$ are in $S$ . for all positive integers $n>1$ , prove that there exists a good set $S$ such that $n \not \in S$ .

Proposed by Seyed Reza Hosseini Dolatabadi