Bertrand postulate - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T21:20:47Z http://mathoverflow.net/feeds/question/2713 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2713/bertrand-postulate Bertrand postulate Ilya Nikokoshev 2009-10-26T23:13:23Z 2010-05-29T02:42:03Z <p>I believe there was an old conjecture that there's <strong>always a prime number between <code>N</code> and <code>2N</code></strong>.</p> <p>What's the history and how is this proven is the easiest/elementary/deepest ways?</p> http://mathoverflow.net/questions/2713/bertrand-postulate/2717#2717 Answer by Michael Lugo for Bertrand postulate Michael Lugo 2009-10-26T23:17:42Z 2009-10-26T23:17:42Z <p>Proven. This is called Bertrand's postulate. Here is <a href="http://www.nd.edu/~dgalvin1/pdf/bertrand.pdf" rel="nofollow">Erdos' elementary proof</a>; the original proof is due to Chebyshev.</p> http://mathoverflow.net/questions/2713/bertrand-postulate/2733#2733 Answer by Ricardo for Bertrand postulate Ricardo 2009-10-27T00:48:29Z 2009-10-27T00:48:29Z <p>Erdös's proof is also contained in Chapter 2 of <a href="http://books.google.com.mx/books?id=KvQr9l0wgf8C" rel="nofollow">Proofs from THE BOOK</a>, by Aigner &amp; Ziegler.</p> http://mathoverflow.net/questions/2713/bertrand-postulate/2858#2858 Answer by engelbrekt for Bertrand postulate engelbrekt 2009-10-27T18:13:55Z 2009-10-27T19:26:51Z <p>There are various proofs of Bertrand's postulate. There is quite an easy one available if one treats it together with the proof of the usual (double) Chebyshev bound as a unit. One optimizes the proof of the Chebyshev bound for subsequently proving Bertrand's Postulate by the method of Ramanujan (getting rid of the appeal to Stirling's formula at the same time).</p> <p>The history of Bertrand's Postulate is set forth in The Development of Prime Number Theory by Wladyslaw Narkiewicz.</p> <p>A comment to the comment by Michael Lugo. The Prime Number Theorem is considerably harder to prove than Bertrand's Postulate, and getting the PNT in the form of good explicit inequalities is hard work on top of that (such inequalities exist, and are useful for some purposes).</p> http://mathoverflow.net/questions/2713/bertrand-postulate/4588#4588 Answer by J. H. S. for Bertrand postulate J. H. S. 2009-11-08T04:02:52Z 2010-05-29T02:42:03Z <p>Note that we can also give a conditional proof of Bertrand's Postulate assuming the veracity of Goldbach's Conjecture. This was the subject matter of a short note that appeared in the sixth issue of volume #112 of the Monthly.</p> <p>Also, it has to be noted that the reference given by Michael Lugo doesn't contain the original proof of Erdős. The original one is to be found <a href="http://www.math-inst.hu/~p_erdos/1932-01.pdf" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/2713/bertrand-postulate/4655#4655 Answer by Kristal Cantwell for Bertrand postulate Kristal Cantwell 2009-11-08T17:56:09Z 2009-11-08T17:56:09Z <p>For large enough x, x+x^.525 contains a prime see:</p> <p>R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II, Proceedings of the London Mathematical Society 83, (2001), 532–562.</p> <p>For more on related results on prime gaps see the follwoing</p> <p><a href="http://en.wikipedia.org/wiki/Prime%5Fgap" rel="nofollow">http://en.wikipedia.org/wiki/Prime_gap</a></p> http://mathoverflow.net/questions/2713/bertrand-postulate/10530#10530 Answer by Anweshi for Bertrand postulate Anweshi 2010-01-02T21:42:00Z 2010-01-02T21:42:00Z <p>See Chandrasekharan, Analyic Number Theory, for the proof by S. S. Pillai. It is quite easy.</p> http://mathoverflow.net/questions/2713/bertrand-postulate/10546#10546 Answer by Ben Weiss for Bertrand postulate Ben Weiss 2010-01-02T23:04:46Z 2010-01-02T23:04:46Z <p>One remark to relate Bertrand's postulate to the prime number theorem: Chebyshev's work was related to bounding ratios of factorials--in particular $\frac{2n!}{n!n!}.$</p> <p>His later proof that $C\frac{x}{\log x} &lt; \pi(x) &lt; D\frac{x}{\log x}$ made use of other ratios (in this case $\frac{30n!n!}{15n!5n!3n!}$). In theory one could try and improve the numbers used in the ratio to asymptotically prove the prime number theorem. Jonathan Bober (among others) have worked on this. He has catalogued many different combinations of ratios of factorials (this also ends up tying into G and E functions....but I'm already out of my depth of what I'm capable of explaining).</p>