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



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



Approximating integers with prime quotients
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):


1. P. 165 in: W. Sierpiński, *Elementary Theory of Numbers*, North-Holland Mathematical Library **31**, North-Holland, Amsterdam, 1988 (2nd edition).
2. Theorem 4 in: P. Ribenboim, *The Book of Prime Number Records*, Springer-Verlag, New York, 1989 (2nd edition).
3. Theorem 4 in: D. Hobby and D. M. Silberger, *Quotients of primes*, Amer. Math. Monthly **100** (1993), No. 1, 50–52 ([click](http://www.jstor.org/stable/2324814?seq=1#page_scan_tab_contents)).
4. Corollary 2 in: P. Starni, *Answers to two questions concerning quotients of primes*, Amer. Math. Monthly **102** (1995), No. 4, 347–349 ([click](http://www.jstor.org/stable/2974957?seq=1#page_scan_tab_contents)).
5. Exercise 4.19 in: B. Fine and G. Rosenberger, *Number Theory: An Introduction via the Distribution of Primes*, Birkhäuser, Boston, 2007.
6. Exercise 218 in: J.-M. De Koninck and A. Mercier, *1001 Problems in Classical Number Theory*, American Mathematical Society, Providence, RI, 2007.
7. Exercise 7, p. 107 in: P. Pollack, *Not Always Buried Deep: A Second Course in Elementary Number Theory*, American Mathematical Society, Providence, RI, 2009.
8. 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](http://www.jstor.org/stable/10.4169/amer.math.monthly.118.08.704)).


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.


