Is this statement true for all positive integers $n\in\mathbb{N}$?
For all $\varepsilon >0$ there are prime numbers $p,q$ such that $|\frac{p}{q} - n| < \varepsilon$.
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asked Sep 22, 2015 at 7:20
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$\begingroup$ Yes, and it is even true that for every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p, q \in \mathbf Z$ such that $|\alpha - p/q| < \varepsilon$; see, e.g., at the bottom of p. 165 in W. Sierpiński, Elementary Theory of Numbers, Amsterdam: North-Holland Mathematical Library 31, North-Holland, 1988 (2nd edition). $\endgroup$– Salvo TringaliCommented Sep 22, 2015 at 7:33
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$\begingroup$ Thanks Salvo! I guess I should delete this question... Or you can put your comment as an answer ... as you wish! $\endgroup$– Dominic van der ZypenCommented Sep 22, 2015 at 7:42
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3$\begingroup$ Some posts on Math.SE: Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$., The set of rational numbers of the form p/p', where p and p' are prime, is dense in $[0, \infty)$ and Are fractions with prime numerator and denominator dense?. Related MO post: Using Quotient of Prime Numbers to Approximation Reals. $\endgroup$– Martin SleziakCommented Sep 22, 2015 at 9:40
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$\begingroup$ You can also read Quotients of Primes by David Hobby and D. M. Silberger. $\endgroup$– WatsonCommented Jan 7, 2017 at 21:07
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$\begingroup$ @Watson Hobby and Silberger's paper is cited in my answer below (item 3 in the list there). $\endgroup$– Salvo TringaliCommented Aug 9, 2017 at 7:57
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2 Answers
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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.
answered Sep 22, 2015 at 7:51
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1$\begingroup$ Possibly Stephan Ramon Garcia, Quotients of Gaussian primes, 2013, later appearing in Amer. Math. Monthly $\endgroup$– HenryCommented Sep 22, 2015 at 11:02
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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?
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$\begingroup$ The best known $\epsilon$ is $0.525$, see Baker-Harman-Pintz: The difference between consecutive primes, II, Proc. London Math. Soc. (3) 83 (2001) 532-562. $\endgroup$ Commented Sep 22, 2015 at 18:18