CIIM 2017 Problem 5

Let $\mathcal{S}$ be a set of integers. Given a real positive $r$, we say that $\mathcal{S}$ is a $r$ - discerning, if for any pair $m, n > 1$ of distinct integers such that $\left| \frac{m-n}{m+n} \right| < r$, there exists $a \in \mathcal{S}$ and $k \geq 1$ such that $a^k$ divides $m$ but not $n$, or $a^k$ divides $n$ but not $m$

1. Show that for every $r > 0$ every $r$ - discerning set contains an infinite number of primes.

2. For every $r > 0$ determine the maximal possible cardinality of $\mathcal{P} \backslash \mathcal{S}$ where $\mathcal{P}$ is the set of primes and $\mathcal{S} \subseteq \mathcal{P}$ is a $r$ - discerning set.