Question

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?

Answer 1

Question 1: The set is dense.

Suppose that we are given a fixed $x\in\mathbb{R}$. Then let $p$ be a large prime. If $p$ is sufficiently large, then there will be a prime $$q\in\left[px,\ px+\left(px\right)^{0.525}\right]$$ by the work of Baker, Harman and Pintz on prime gaps. This implies that $$\left|x-\frac{q}{p}\right|\ll_x p^{-0.475},$$ which becomes arbitrarily small as we take $p\rightarrow\infty $. This proves that for any $\epsilon>0$, there exists $p,q$ such that $\left|x-\frac{q}{p}\right|\leq \epsilon.$

Question 2: We can find infinitely many solutions to $$1\leq qp-rs\leq a$$ for primes $p,q,r,s$ and all $a\geq 26$. Under the Elliott-Halberstam Conjecture, we can take $a\geq 6$.

This is a corollary of the work of Goldston, Graham, Pintz and Yıldırım on the gaps between almost primes. They prove that if $q_n$ is the $n^{th}$ almost prime, then $$\liminf_{n\rightarrow \infty} q_{n+1}-q_n \leq 26,$$ and that the upper bound may be reduced to $6$ under the Elliott-Halberstam Conjecture. Since $q_n=pq$ and $q_{n+1}=rs$ where $p,q,r,s$ are primes, this yields the above claim.

Edit: The more recent work of Goldston, Graham, Pintz and Yıldırım show that we can take $a=6$ unconditionally. (Thank you to quid for mentioning this in the comments)

Answer 2

Here is a simpler solution to question 1. By the prime number theorem, the $n$th prime $p_n$ admits the asymptotic estimate $p_n \sim n\log n$. It follows for any real number $x>0$ that $p_{[nx]}/p_n \rightarrow x$ as $n \rightarrow \infty$. The rate of convergence for this explicit ratio of primes tending to $x$ is pretty slow, however. For instance, taking $x = \pi$, I checked with PARI that $p_{[n\pi]}/p_n$ is $3.642$ when $n = 1000$ and $3.517$ when $n=10000$.

Answer 3

Question 1 comes up a lot. For example, it was discussed on sci.math back in 1990. It was also discussed in print in the following article:

Quotients of primes, by David Hobby and D. M. Silberger, Amer. Math Monthly 100 (1993), 50–52.

More recently it has shown up on Yahoo Answers and math.stackexchange.com.