Using Quotient of Prime Numbers to Approximation Reals
We know a positive rational number can be uniquely written as $m/n$ where $m$ and $n$ are coprime positive integers. Particularly, we can pick out those numbers with $m$ and $n$ both prime.
Question 1: Is the collection of all such numbers dense on the positive half of the real line?
Furthermore, we can ask about the efficiency of approximation, more precisely:
Question 2: Suppose we have an inequality $1\le ps-qr\le a$. Fix some $a$, can we find infinitely many solutions where $p$,$s,$,$q$,$r$ are positive primes?