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Reddwarf2956 (talk | contribs) Undid revision 746019377 by 146.196.34.184 (talk) No grammar error, that statement is correct with 2*n. |
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{{short description|Prime fulfilling an inequality related to the prime-counting function}}
{{Distinguish|Hardy–Ramanujan number}}
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In 1919, Ramanujan published a new proof of [[Bertrand's postulate]] which, as he notes, was first proved by [[Pafnuty Chebyshev|Chebyshev]].<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> At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:
: <math>\pi(x) - \pi\left( \frac x
where <math>\pi(x)</math> is the [[prime-counting function]], equal to the number of primes less than or equal to ''x''.
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The converse of this result is the definition of Ramanujan primes:
:The ''n''th Ramanujan prime is the least integer ''R<sub>n</sub>'' for which <math>\pi(x) - \pi(x/2) \ge n,</math> for all ''x'' ≥ ''R<sub>n</sub>''.<ref>{{MathWorld|
The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.
Note that the integer ''R<sub>n</sub>'' is necessarily a prime number: <math>\pi(x) - \pi(x/2)</math> and, hence, <math>\pi(x)</math> must increase by obtaining another prime at ''x'' = ''R<sub>n</sub>''. Since <math>\pi(x) - \pi(x/2)</math> can increase by at most 1,
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:''R''<sub>''n''</sub> ~ ''p''<sub>2''n''</sub> (''n'' → ∞).
All these results were proved by Sondow (2009),<ref>{{Citation |first=J. |last=Sondow |title=Ramanujan primes and Bertrand's postulate |journal=Amer. Math. Monthly |volume=116 |issue=7 |year=2009 |pages=630–635 |arxiv=0907.5232 |doi=10.4169/193009709x458609}}</ref> except for the upper bound ''R''<sub>''n''</sub> < ''p''<sub>3''n''</sub> which was conjectured by him and proved by Laishram (2010).<ref>{{Citation |first=S. |last=Laishram |title=On a conjecture on Ramanujan primes |journal=[[International Journal of Number Theory]] |volume=6 |issue=8 |year=2010 |pages=1869–1873 |url=http://www.isid.ac.in/~shanta/PAPERS/RamanujanPrimes-IJNT.pdf |doi=10.1142/s1793042110003848|citeseerx=10.1.1.639.4934 }}.</ref> The bound was improved by Sondow, Nicholson, and Noe (2011)<ref>{{Citation |first1=J. |last1=Sondow |first2=J. |last2=Nicholson |first3=T.D. |last3=Noe |title=Ramanujan primes: bounds, runs, twins, and gaps |journal=Journal of Integer Sequences |volume=14|year=2011 |pages=11.6.2|url=http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Noe/noe12.pdf|arxiv=1105.2249|bibcode=2011arXiv1105.2249S }}</ref> to
:<math>R_n \le \frac{41}{47} \ p_{3n}</math>
which is the optimal form of ''R''<sub>''n''</sub> ≤ ''c·p''<sub>3''n''</sub> since it is an equality for ''n'' = 5.
==References==
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