Approximating integers with prime quotients Is this statement true for all positive integers $n\in\mathbb{N}$?


For all $\varepsilon >0$ there are prime numbers $p,q$ such that $|\frac{p}{q} - n| < \varepsilon$.


 A: It is known that there exists a prime between $n$ and $n+n^{\epsilon}$ for $n$ large enough, where the best known $\epsilon$ is 0.525. Therefore, given a large enough prime $q$, choose a prime $p$ between $q\alpha$ and $q\alpha+(q\alpha)^{\epsilon}$ to get 
$$|\alpha-p/q|<Cq^{\epsilon-1},$$
where $C$ depends only on $\alpha$. Assuming the Reimann Hypothesis, this bound can be improved to 
$$|\alpha-p/q|<Cq^{-1/2}\log q,$$
for infinitely many pairs of primes $(p,q)$. With recent results on prime gaps, this bound is extremely better when $\alpha=1$. One can then ask this question:
${\bf Question:}$ Let $\alpha$ be a positive real number. Does there always exist infinitely many pairs of primes $(p,q)$ such that
$$|\alpha-p/q|<C/q,$$
where $C$ is a universal constant? 
A: This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every  $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is a (partial) list of references where it is mentioned or proved (to try to convince you that yes, it is really well-known):


*

*P. 165 in: W. Sierpiński, Elementary Theory of Numbers, North-Holland Mathematical Library 31, North-Holland, Amsterdam, 1988 (2nd edition).

*Theorem 4 in: P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989 (2nd edition).

*Theorem 4 in: D. Hobby and D. M. Silberger, Quotients of primes, Amer. Math. Monthly 100 (1993), No. 1, 50–52 (click).

*Corollary 2 in:  P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Monthly 102 (1995), No. 4, 347–349 (click).

*Exercise 4.19 in: B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, Boston, 2007.

*Exercise 218 in: J.-M. De Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, RI, 2007.

*Exercise 7, p. 107 in: P. Pollack, Not Always Buried Deep:  A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009.

*Corollary 5 in: S. R. Garcia, V. Selhorst-Jones, D. E. Poore, and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Monthly 118 (2011), No. 8, 704–711 (click).


For the record: I compiled this list at some point, based on information drawn from an arXiv preprint (unfortunately, I didn't take note of either a link, the author(s), or the title of the preprint, so I don't know how to find it again), as the question popped up in relation to a certain property of densities.
