Are there any interesting or lesser known proofs related to Bertrand's Postulate - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:06:40Z http://mathoverflow.net/feeds/question/103736 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103736/are-there-any-interesting-or-lesser-known-proofs-related-to-bertrands-postulate Are there any interesting or lesser known proofs related to Bertrand's Postulate Larry Freeman 2012-08-01T22:54:28Z 2013-05-03T11:46:59Z <p>There are 3 standard proofs of Bertrand's Postulate:</p> <p>(1) Chebyshev's original proof</p> <p>(2) Ramanujan's simplification of Chebyshev's proof</p> <p>(3) Erdos's proof</p> <p>I recently learned about the very simple proof that if the Goldbach conjecture is true, then Bertrand's postulate follows (see <a href="http://www.proofwiki.org/wiki/Goldbach_implies_Bertrand" rel="nofollow">here</a>).</p> <p>Does anyone know of any other proofs? There are recent proofs that extend Bertrand's postulate to show that there is always a prime in $2n$/$3n$ and $3n$/$4n$. </p> <p>I am wondering if there aren't other lesser known proofs that take a different approach to establish the existence of a prime between $n$ and $2n$.</p> <p>Thanks,</p> <p>-Larry</p> http://mathoverflow.net/questions/103736/are-there-any-interesting-or-lesser-known-proofs-related-to-bertrands-postulate/103744#103744 Answer by Benjamin Dickman for Are there any interesting or lesser known proofs related to Bertrand's Postulate Benjamin Dickman 2012-08-02T00:35:48Z 2012-08-02T00:35:48Z <p>Bertrand's Postulate follows as a direct consequence of the following theorem of J.J. Sylvester: </p> <p><strong>Theorem (Sylvester, 1892):</strong> Let $k$ be a positive integer. Then at least one of any $k$ consecutive integers greater than $k$ is divisible by a prime greater than $k$.</p> <p>(For comparison: Chebyshev's analytic proof dates to 1850; Erdos' elementary proof dates to 1932.)</p> <p>See Theorem 6 (p. 6) in <a href="http://www.math.sc.edu/~filaseta/papers/schurpaper.pdf" rel="nofollow">http://www.math.sc.edu/~filaseta/papers/schurpaper.pdf</a>, from which I quote: </p> <p>"The theorem implies immediately that for any positive integer $k$, one of $k+1, k+2, \ldots, 2k$ is a prime (since one of these integers must be divisible by a prime $\geq k+1).$"</p> <p>Unfortunately, my internet search has not led me to the original paper by Sylvester.</p> http://mathoverflow.net/questions/103736/are-there-any-interesting-or-lesser-known-proofs-related-to-bertrands-postulate/103745#103745 Answer by Igor Rivin for Are there any interesting or lesser known proofs related to Bertrand's Postulate Igor Rivin 2012-08-02T00:38:14Z 2012-08-07T13:26:26Z <p>A stronger version is proved by Jonathan Sondow in <a href="http://arxiv.org/abs/0907.5232" rel="nofollow">this arxiv preprint</a> (which looks like a monthly paper).</p> http://mathoverflow.net/questions/103736/are-there-any-interesting-or-lesser-known-proofs-related-to-bertrands-postulate/104176#104176 Answer by unknown (google) for Are there any interesting or lesser known proofs related to Bertrand's Postulate unknown (google) 2012-08-07T06:46:23Z 2012-08-07T06:46:23Z <p>Nice question! For a point of view from the perspective of Goldbach's conjecture, perhaps one can consider also Theorem 3.7 of "The Hardy-Littlewood Method", 2nd edition, by R.C. Vaughan.</p> http://mathoverflow.net/questions/103736/are-there-any-interesting-or-lesser-known-proofs-related-to-bertrands-postulate/104236#104236 Answer by Aaron Meyerowitz for Are there any interesting or lesser known proofs related to Bertrand's Postulate Aaron Meyerowitz 2012-08-07T21:34:47Z 2012-08-07T21:34:47Z <p>The <a href="http://www.m-hikari.com/ijcms-password/ijcms-password13-16-2006/elbachraouiIJCMS13-16-2006.pdf" rel="nofollow">proofs</a> for $(2n,3n)$ and $(3n,4n)$ are elementary and very pleasing (based on a quick look.) It is <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.pja/1195570997" rel="nofollow">known</a> that there is always a prime between $k$ and $\frac{6k}{5}\$ for $k \gt 24.$ The proof is more involved but do not use anything analytic. Putting $k=2n$ etc gives a prime in $(4n,\frac{24n}{5})$ (except $n=1,2,6$) and hence in $(4n,5n)$ except for $n=1,2$ and one in $(5n,6n)$ except for $n=1$. </p> http://mathoverflow.net/questions/103736/are-there-any-interesting-or-lesser-known-proofs-related-to-bertrands-postulate/106470#106470 Answer by Larry Freeman for Are there any interesting or lesser known proofs related to Bertrand's Postulate Larry Freeman 2012-09-06T01:05:33Z 2012-09-06T01:05:33Z <p>I found an interesting proof today that demonstrates a stronger form of Bertrand's Postulate. I hadn't seen it before:</p> <p>Abstract. In this paper we give a stronger form of Bertrand's postulate and use it to prove that every positive integer, except 1, 2, 4, 6, and 9, can be written as the sum of distinct odd primes.</p> <p><a href="http://www.ams.org/journals/proc/1972-033-02/S0002-9939-1972-0292746-6/S0002-9939-1972-0292746-6.pdf" rel="nofollow">http://www.ams.org/journals/proc/1972-033-02/S0002-9939-1972-0292746-6/S0002-9939-1972-0292746-6.pdf</a></p> http://mathoverflow.net/questions/103736/are-there-any-interesting-or-lesser-known-proofs-related-to-bertrands-postulate/129521#129521 Answer by Armin for Are there any interesting or lesser known proofs related to Bertrand's Postulate Armin 2013-05-03T11:46:59Z 2013-05-03T11:46:59Z <p>Re-extending Chebyshev’s theorem about Bertrand’s conjecture:</p> <p><a href="http://link.springer.com/article/10.1007%2Fs11253-008-0034-7" rel="nofollow">http://link.springer.com/article/10.1007%2Fs11253-008-0034-7</a></p>