Using Quotient of Prime Numbers to Approximation Reals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:06:52Z http://mathoverflow.net/feeds/question/117191 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117191/using-quotient-of-prime-numbers-to-approximation-reals Using Quotient of Prime Numbers to Approximation Reals Arkus 2012-12-25T08:45:31Z 2012-12-26T00:54:06Z <p>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 <strong>prime</strong>.</p> <p><strong>Question 1</strong>: Is the collection of all such numbers dense on the positive half of the real line?</p> <p>Furthermore, we can ask about the efficiency of approximation, more precisely:</p> <p><strong>Question 2</strong>: 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?</p> http://mathoverflow.net/questions/117191/using-quotient-of-prime-numbers-to-approximation-reals/117192#117192 Answer by Eric Naslund for Using Quotient of Prime Numbers to Approximation Reals Eric Naslund 2012-12-25T09:02:17Z 2012-12-25T17:50:05Z <p><strong>Question 1:</strong> The set is dense. </p> <p>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 <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf" rel="nofollow">the work</a> of Baker, Harman and Pintz on <a href="http://en.wikipedia.org/wiki/Prime_gap" rel="nofollow">prime gaps</a>. 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 $\left|x-\frac{q}{p}\right|\leq \epsilon.$</p> <p><strong>Question 2:</strong> 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 <a href="http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture" rel="nofollow">Elliott-Halberstam Conjecture</a>, we can take $a\geq 6$.</p> <p>This is a corollary of <a href="http://arxiv.org/abs/math/0506067" rel="nofollow">the work</a> 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.</p> <p><strong>Edit:</strong> The <a href="http://arxiv.org/abs/math/0609615" rel="nofollow">more recent work</a> of Goldston, Graham, Pintz and Yıldırım show that we can take $a=6$ unconditionally. (Thank you to quid for mentioning this in the comments)</p> http://mathoverflow.net/questions/117191/using-quotient-of-prime-numbers-to-approximation-reals/117207#117207 Answer by KConrad for Using Quotient of Prime Numbers to Approximation Reals KConrad 2012-12-25T18:45:59Z 2012-12-25T18:45:59Z <p>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 $n=10000$.</p> http://mathoverflow.net/questions/117191/using-quotient-of-prime-numbers-to-approximation-reals/117219#117219 Answer by Timothy Chow for Using Quotient of Prime Numbers to Approximation Reals Timothy Chow 2012-12-26T00:54:06Z 2012-12-26T00:54:06Z <p>Question 1 comes up a lot. For example, it was discussed on <a href="https://groups.google.com/forum/?fromgroups=#!msg/sci.math/LWOtoA-w7XA/rPeoqgalh10J" rel="nofollow">sci.math</a> back in 1990. It was also discussed in print in the following article:</p> <blockquote> <p>Quotients of primes, by David Hobby and D. M. Silberger, <i>Amer. Math Monthly</i> <b>100</b> (1993), 50&ndash;52.</p> </blockquote> <p>More recently it has shown up on <a href="http://answers.yahoo.com/question/index?qid=20081022093608AAZkzbJ" rel="nofollow">Yahoo Answers</a> and <a href="http://math.stackexchange.com/questions/4066/rationals-of-the-form-fracpq-where-p-q-are-primes-in-a-b" rel="nofollow">math.stackexchange.com</a>.</p>