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en>BG19bot: WP:CHECKWIKI error fix. Broken bracket. Do general fixes if a problem exists. - using AWB (10480)

2014-11-11T00:22:49Z

WP:CHECKWIKI error fix. Broken bracket. Do general fixes if a problem exists. - using AWB (10480)

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	~~{{Distinguish\|Hardy–Ramanujan number}}~~		The author is known as Wilber Pegues. North Carolina is the place he enjoys most but now he is contemplating other options. My working day job is an information officer but tarot readings - [http://www.seekavideo.com/playlist/2199/video/ www.seekavideo.com], I've already utilized for an additional one. What me and my family members adore is bungee jumping but I've been taking on new things recently.<br><br>Also visit my blog post - online [http://m-card.co.kr/xe/mcard_2013_promote01/29877 real psychics] ([http://www.article-galaxy.com/profile.php?a=143251 click the next internet site])

	~~In [[mathematics]], a '''Ramanujan prime'''~~ is ~~a [[prime number]] that satisfies a result proven by [[Srinivasa Ramanujan]] relating to the [[prime-counting function]].~~

	~~==Origins and definition==~~
	~~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(x/2)</math> ≥ 1, 2, 3, 4, 5, ... for all ''x'' ≥ 2, 11, 17, 29, 41, ... {{OEIS2C\|id=A104272}} respectively,~~

	~~where <math>\pi(x)</math>~~ is ~~the [[prime-counting function]], equal to the number of primes less than or equal to ''x''~~.

	~~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)</math> ≥ ''n'', for all~~ ~~''x'' ≥ ''R<sub>n</sub>''.<ref>{{MathWorld\|\|authorlink=Jonathan Sondow\|author=Jonathan Sondow\|title=Ramanujan Prime\|urlname=RamanujanPrime}}</ref>~~

	~~The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Equivalently,~~

	~~:Ramanujan primes are the least integers ''R<sub>n</sub>'' for which there are at least ''n'' primes between ''x'' and ''x''/2 for all ''x'' ≥ ''R<sub>n</sub>''.~~

	~~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,~~

	~~: <math>\pi(</math>''R<sub>n</sub>''<math>) - \pi(</math>''R<sub>n</sub>''<math>/2) = n</math>.~~

	~~==Bounds and an asymptotic formula==~~

	~~For all ''n'' ≥ 1, the bounds~~

	~~:2''n'' ln 2''n'' < ''R<sub>n</sub>'' < 4''n'' ln 4''n''~~

	~~hold. If ''n'' > 1, then also~~

	~~:''p''<sub>2''n''</sub> < ''R<sub>n</sub>'' < ''p''<sub>3''n''</sub>~~

	~~where ''p''<sub>''n''</sub> is the ''n''th prime number.~~

	~~As ''n'' tends to infinity, ''R''<sub>''n''</sub> is~~ [~~[Asymptotic analysis\|asymptotic]] to the 2''n''th prime, i.e.,~~

	~~:''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 \|year=2009 \|pages=630–635 \|arxiv=0907.5232 }}</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\|year=2010 \|pages=1869–1873\|url=http://www.~~isid~~.~~ac.in~~/~~~shanta~~/PAPERS/RamanujanPrimes.pdf}}.</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}}</ref> to~~

	~~:<math>R_n \le \frac{41}{47} \ p_{3n}</math>~~

	~~which is optimal since it is an equality for ''n'' = 5.~~

	~~== Generalized Ramanujan primes ==~~

	~~Given a constant ''c'' between 0 and 1, the ''n''th ''c''-Ramanujan prime is defined as the~~
	~~smallest integer ''R<sub>c~~,~~n</sub>~~'~~' with the property that~~ for ~~any integer ''x ≥ R<sub>c,n</sub>'' there are at least ''n'' primes between ''cx''~~
	and ~~''x'', that~~ is~~, <math>\pi(x) - \pi(cx) \ge n</math>. In particular, when~~ '~~'c'' = 1/2, the ''n''th 1/2-Ramanujan prime is equal to the ''n''th Ramanujan prime: ''R''<sub>0~~.~~5,''n''~~<~~/sub~~> ~~= ''R~~<~~sub~~>~~n</sub>''.~~

	~~For ''c'' = 1/4 and 3/4, the sequence of ''c''~~-~~Ramanujan primes begins~~

	:~~''R''<sub>0.25,''n''<~~/~~sub> = 2, 3, 5, 13, 17, ... {{OEIS2C\|id=A193761}},~~

	~~:''R''<sub>0.75,''n''<~~/~~sub> = 11, 29, 59, 67, 101, ... {{OEIS2C\|id=A193880}}~~.

	It is known<ref>{{Citation \|first1=N. \|last1=Amersi \|first2=O. \|last2=Beckwith \|first3=S.J. \|last3=Miller \|first4=R. \|last4=Ronan \|first5=J. \|last5=Sondow \|title=Generalized Ramanujan primes \|year=2011 \|arxiv=1108.0475}}</ref> that, for all ''n'' and ''c'', the ''n''th ''c''-Ramanujan prime ''R<sub>c,n</sub>'' exists and is indeed prime. ~~Also, as ''n'' tends to infinity, ''R<sub>c,n<~~/~~sub>'' is asymptotic to ''p''<sub>''n''~~/~~(1 − ''c'')</sub>~~

	~~:''R''<sub>''c'',''n''</sub> ~ ''p''<sub>''n''/(1 − ''c'')</sub> (''n'' → ∞)~~

	~~where ''p''<sub>''n''~~/(~~1 − ''c'')</sub> is the <math>\lfloor</math>''n''/(1 − ''c'')<math>\rfloor </math>th prime and <math>\lfloor .\rfloor</math> is the~~ [~~[Floor and ceiling functions\|floor]] function.~~

	~~== Ramanujan prime corollary ==~~

	:~~<math>2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\, ,<~~/~~math>~~

	~~i.e. ''p''<sub>''k''<~~/~~sub> is the ''k''th prime and the ''n''th Ramanujan prime~~.

	~~This is very useful in showing the number of primes in the range [p<sub>k</sub>, 2*p<sub>i~~-~~n</sub>] is greater than or equal to 1~~. ~~By taking into account the size of the gaps between primes in [''p''<sub>''i''−''n''</sub>,''p''<sub>''k''<~~/~~sub>], one can see that the average prime gap is about ln(''p''<sub>''k''</sub>) using the following ''R''<sub>''n''</sub>/(2''n'') ~ ln(''R''<sub>''n''</sub>)~~.

	~~Proof of Corollary:~~

	~~If ''p''<sub>''i''</sub> > ''R''<sub>''n''</sub>, then ''p''<sub>''i''</sub> is odd and ''p''<sub>''i''</sub> − 1 ≥ ''R''<sub>''n''</sub>, and hence~~
	~~''π''(''p''<sub>''i''</sub> − 1) − ''π''(''p''<sub>''i''</sub>/2) = ''π''(''p''<sub>''i''</sub> − 1) − ''π''((''p''<sub>''i''</sub> − 1)/2) ≥ ''n''.~~
	Thus ''p''<sub>''i''</sub> − 1 ≥ ''p''<sub>''i''−1</sub> > ''p''<sub>''i''−2</sub> > ''p''<sub>''i''−3</sub> > ... > ''p''<sub>''i''−''n''</sub> > ''p''<sub>''i''</sub>/2, and so 2''p''<sub>''i''−''n''</sub> > ''p''<sub>''i''</sub>.

	~~An example of this corollary:~~

	~~With ''n'' = 1000, ''R''<sub>''n''</sub> = ''p''<sub>''k''</sub> = 19403, and ''k''~~ = ~~2197, therefore ''i'' ≥ 2198 and ''i''−''n'' ≥ 1198.~~
	The smallest ''i'' − ''n'' prime is ''p''<sub>''i''−''n''</sub> = 9719, therefore 2''p''<sub>''i''−''n''</sub> = 2 × 9719 = 19438. The 2198th prime, ''p''<sub>''i''</sub>, is between ''p''<sub>''k''</sub> = 19403 and 2''p''<sub>''i''−''n''</sub> = 19438 and is 19417.

	~~The left side of~~ the ~~Ramanujan Prime Corollary is the {{OEIS2C\|id=A168421}}; the right side is the {{OEIS2C\|id=A168425}}.~~
	~~The values of <math>\pi(R_n~~)~~\,</math> are in the {{OEIS2C\|id=A179196}}.~~

	~~The Ramanujan Prime Corollary is due to John Nicholson.~~

	~~==References==~~
	~~{{Reflist}}~~

	~~{{Prime number classes}}~~

	~~[[Category:Srinivasa Ramanujan]]~~
	~~[[Category:Classes of prime numbers]]~~

en>Yobot: WP:CHECKWIKI error fixes using AWB (9345)

2013-07-11T10:08:54Z

WP:CHECKWIKI error fixes using AWB (9345)

← Older revision		Revision as of 12:08, 11 July 2013
Line 1:		Line 1:
	~~Wilber Berryhill~~ is ~~what his spouse enjoys~~ to ~~contact him~~ and he ~~totally loves~~ this title. ~~Distributing production~~ is where ~~her main income arrives from~~. ~~I am really fond of~~ to go to ~~karaoke but I~~'~~ve been taking~~ on ~~new issues recently~~. ~~Some time ago she selected to reside~~ in ~~Alaska~~ and ~~her parents live close by~~.<br><br>~~Feel free~~ to ~~visit my blog post~~ :: [~~http~~://~~galab~~-~~Work~~.Cs.~~pusan~~.ac.kr/~~Sol09B~~/~~?document_srl~~=~~1489804 love psychic readings~~]		{{Distinguish\|Hardy–Ramanujan number}}

			In [[mathematics]], a '''Ramanujan prime''' is a [[prime number]] that satisfies a result proven by [[Srinivasa Ramanujan]] relating to the [[prime-counting function]].

			==Origins and definition==
			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(x/2)</math> ≥ 1, 2, 3, 4, 5, ... for all ''x'' ≥ 2, 11, 17, 29, 41, ... {{OEIS2C\|id=A104272}} respectively,

			where <math>\pi(x)</math> is the [[prime-counting function]], equal to the number of primes less than or equal to ''x''.

			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)</math> ≥ ''n'', for all ''x'' ≥ ''R<sub>n</sub>''.<ref>{{MathWorld\|\|authorlink=Jonathan Sondow\|author=Jonathan Sondow\|title=Ramanujan Prime\|urlname=RamanujanPrime}}</ref>

			The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Equivalently,

			:Ramanujan primes are the least integers ''R<sub>n</sub>'' for which there are at least ''n'' primes between ''x'' and ''x''/2 for all ''x'' ≥ ''R<sub>n</sub>''.

			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,

			: <math>\pi(</math>''R<sub>n</sub>''<math>) - \pi(</math>''R<sub>n</sub>''<math>/2) = n</math>.

			==Bounds and an asymptotic formula==

			For all ''n'' ≥ 1, the bounds

			:2''n'' ln 2''n'' < ''R<sub>n</sub>'' < 4''n'' ln 4''n''

			hold. If ''n'' > 1, then also

			:''p''<sub>2''n''</sub> < ''R<sub>n</sub>'' < ''p''<sub>3''n''</sub>

			where ''p''<sub>''n''</sub> is the ''n''th prime number.

			As ''n'' tends to infinity, ''R''<sub>''n''</sub> is [[Asymptotic analysis\|asymptotic]] to the 2''n''th prime, i.e.,

			:''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 \|year=2009 \|pages=630–635 \|arxiv=0907.5232 }}</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\|year=2010 \|pages=1869–1873\|url=http://www.isid.ac.in/~shanta/PAPERS/RamanujanPrimes.pdf}}.</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}}</ref> to

			:<math>R_n \le \frac{41}{47} \ p_{3n}</math>

			which is optimal since it is an equality for ''n'' = 5.

			== Generalized Ramanujan primes ==

			Given a constant ''c'' between 0 and 1, the ''n''th ''c''-Ramanujan prime is defined as the
			smallest integer ''R<sub>c,n</sub>'' with the property that for any integer ''x ≥ R<sub>c,n</sub>'' there are at least ''n'' primes between ''cx''
			and ''x'', that is, <math>\pi(x) - \pi(cx) \ge n</math>. In particular, when ''c'' = 1/2, the ''n''th 1/2-Ramanujan prime is equal to the ''n''th Ramanujan prime: ''R''<sub>0.5,''n''</sub> = ''R<sub>n</sub>''.

			For ''c'' = 1/4 and 3/4, the sequence of ''c''-Ramanujan primes begins

			:''R''<sub>0.25,''n''</sub> = 2, 3, 5, 13, 17, ... {{OEIS2C\|id=A193761}},

			:''R''<sub>0.75,''n''</sub> = 11, 29, 59, 67, 101, ... {{OEIS2C\|id=A193880}}.

			It is known<ref>{{Citation \|first1=N. \|last1=Amersi \|first2=O. \|last2=Beckwith \|first3=S.J. \|last3=Miller \|first4=R. \|last4=Ronan \|first5=J. \|last5=Sondow \|title=Generalized Ramanujan primes \|year=2011 \|arxiv=1108.0475}}</ref> that, for all ''n'' and ''c'', the ''n''th ''c''-Ramanujan prime ''R<sub>c,n</sub>'' exists and is indeed prime. Also, as ''n'' tends to infinity, ''R<sub>c,n</sub>'' is asymptotic to ''p''<sub>''n''/(1 − ''c'')</sub>

			:''R''<sub>''c'',''n''</sub> ~ ''p''<sub>''n''/(1 − ''c'')</sub> (''n'' → ∞)

			where ''p''<sub>''n''/(1 − ''c'')</sub> is the <math>\lfloor</math>''n''/(1 − ''c'')<math>\rfloor </math>th prime and <math>\lfloor .\rfloor</math> is the [[Floor and ceiling functions\|floor]] function.

			== Ramanujan prime corollary ==

			:<math>2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\, ,</math>

			i.e. ''p''<sub>''k''</sub> is the ''k''th prime and the ''n''th Ramanujan prime.

			This is very useful in showing the number of primes in the range [p<sub>k</sub>, 2*p<sub>i-n</sub>] is greater than or equal to 1. By taking into account the size of the gaps between primes in [''p''<sub>''i''−''n''</sub>,''p''<sub>''k''</sub>], one can see that the average prime gap is about ln(''p''<sub>''k''</sub>) using the following ''R''<sub>''n''</sub>/(2''n'') ~ ln(''R''<sub>''n''</sub>).

			Proof of Corollary:

			If ''p''<sub>''i''</sub> > ''R''<sub>''n''</sub>, then ''p''<sub>''i''</sub> is odd and ''p''<sub>''i''</sub> − 1 ≥ ''R''<sub>''n''</sub>, and hence
			''π''(''p''<sub>''i''</sub> − 1) − ''π''(''p''<sub>''i''</sub>/2) = ''π''(''p''<sub>''i''</sub> − 1) − ''π''((''p''<sub>''i''</sub> − 1)/2) ≥ ''n''.
			Thus ''p''<sub>''i''</sub> − 1 ≥ ''p''<sub>''i''−1</sub> > ''p''<sub>''i''−2</sub> > ''p''<sub>''i''−3</sub> > ... > ''p''<sub>''i''−''n''</sub> > ''p''<sub>''i''</sub>/2, and so 2''p''<sub>''i''−''n''</sub> > ''p''<sub>''i''</sub>.

			An example of this corollary:

			With ''n'' = 1000, ''R''<sub>''n''</sub> = ''p''<sub>''k''</sub> = 19403, and ''k'' = 2197, therefore ''i'' ≥ 2198 and ''i''−''n'' ≥ 1198.
			The smallest ''i'' − ''n'' prime is ''p''<sub>''i''−''n''</sub> = 9719, therefore 2''p''<sub>''i''−''n''</sub> = 2 × 9719 = 19438. The 2198th prime, ''p''<sub>''i''</sub>, is between ''p''<sub>''k''</sub> = 19403 and 2''p''<sub>''i''−''n''</sub> = 19438 and is 19417.

			The left side of the Ramanujan Prime Corollary is the {{OEIS2C\|id=A168421}}; the right side is the {{OEIS2C\|id=A168425}}.
			The values of <math>\pi(R_n)\,</math> are in the {{OEIS2C\|id=A179196}}.

			The Ramanujan Prime Corollary is due to John Nicholson.

			==References==
			{{Reflist}}

			{{Prime number classes}}

			[[Category:Srinivasa Ramanujan]]
			[[Category:Classes of prime numbers]]

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2011-01-13T23:43:30Z

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Wilber Berryhill is what his spouse enjoys to contact him and he totally loves this title. Distributing production is where her main income arrives from. I am really fond of to go to karaoke but I've been taking on new issues recently. Some time ago she selected to reside in Alaska and her parents live close by.<br><br>Feel free to visit my blog post :: [http://galab-Work.Cs.pusan.ac.kr/Sol09B/?document_srl=1489804 love psychic readings]