Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Bertrand's postulate
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Generalizations == In 1919, [[Srinivasa Aaiyangar Ramanujan|Ramanujan]] (1887–1920) used properties of the [[Gamma function]] to give a simpler proof than Chebyshev's.<ref>{{Citation |first=S. |last=Ramanujan |title=A proof of Bertrand's postulate |journal=Journal of the Indian Mathematical Society |volume=11 |year=1919 |pages=181–182 |url=http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper24/page1.htm }}</ref> His short paper included a generalization of the postulate, from which would later arise the concept of [[Ramanujan prime]]s. Further generalizations of Ramanujan primes have also been discovered; for instance, there is a proof that :<math>2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\, ,</math> with ''p''<sub>''k''</sub> the ''k''th prime and ''R''<sub>''n''</sub> the ''n''th Ramanujan prime. Other generalizations of Bertrand's postulate have been obtained using elementary methods. (In the following, ''n'' runs through the set of positive integers.) In 1973, [[Denis Hanson]] proved that there exists a prime between 3''n'' and 4''n''.<ref>{{Citation | title = On a theorem of Sylvester and Schur | last = Hanson | first = Denis | year = 1973 | journal = [[Canadian Mathematical Bulletin]] | volume = 16 | issue = 2 | pages = 195–199| doi = 10.4153/CMB-1973-035-3 | doi-access=free }}.</ref> In 2006, apparently unaware of Hanson's result, [[M. El Bachraoui]] proposed a proof that there exists a prime between 2''n'' and 3''n''.<ref>{{Citation | title = Primes in the interval [2n,3n] | last = El Bachraoui | first = Mohamed | year = 2006 | journal = International Journal of Contemporary Mathematical Sciences| volume = 1 }}</ref> El Bachraoui's proof is an extension of Erdős's arguments for the primes between n and 2n. Shevelev, Greathouse, and Moses (2013) discuss related results for similar intervals.<ref>{{Citation|last=Shevelev | first=Vladimir | last2=Greathouse | first2=Charles R. | last3=Moses | first3=Peter J. C. |title=On Intervals (kn,(k + 1)n) Containing a Prime for All n > 1 |journal=Journal of Integer Sequences|volume=16|issue=7|date=2013|url=https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.pdf|issn=1530-7638}}</ref> Bertrand’s postulate over the Gaussian integers is an extension of the idea of the distribution of primes, but in this case on the complex plane. Thus, as Gaussian primes extend over the plane and not only along a line, and doubling a [[complex number]] is not simply multiplying by 2 but doubling its norm (multiplying by 1+i), different definitions lead to different results, some are still conjectures, some proven.<ref name="Madhuparna Das">{{Citation|title= Generalization of Bertrand’s postulate for Gaussian primes|author=Madhuparna Das|arxiv=1901.07086v2 |year=2019}}</ref>
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)