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| {{Distinguish|Hardy–Ramanujan number}}
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| In [[mathematics]], a '''Ramanujan prime''' is a [[prime number]] that satisfies a result proven by [[Srinivasa Ramanujan]] relating to the [[prime-counting function]].
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| ==Origins and definition==
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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:
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| : <math>\pi(x) - \pi(x/2)</math> ≥ 1, 2, 3, 4, 5, ... for all ''x'' ≥ 2, 11, 17, 29, 41, ... {{OEIS2C|id=A104272}} respectively,
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| 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:
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| :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>
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| The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Equivalently,
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| :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>''.
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| 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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| : <math>\pi(</math>''R<sub>n</sub>''<math>) - \pi(</math>''R<sub>n</sub>''<math>/2) = n</math>.
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| ==Bounds and an asymptotic formula==
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| For all ''n'' ≥ 1, the bounds
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| :2''n'' ln 2''n'' < ''R<sub>n</sub>'' < 4''n'' ln 4''n''
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| hold. If ''n'' > 1, then also
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| :''p''<sub>2''n''</sub> < ''R<sub>n</sub>'' < ''p''<sub>3''n''</sub>
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| where ''p''<sub>''n''</sub> is the ''n''th prime number.
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| As ''n'' tends to infinity, ''R''<sub>''n''</sub> is [[Asymptotic analysis|asymptotic]] to the 2''n''th prime, i.e.,
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| :''R''<sub>''n''</sub> ~ ''p''<sub>2''n''</sub> (''n'' → ∞).
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| 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
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| :<math>R_n \le \frac{41}{47} \ p_{3n}</math>
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| which is optimal since it is an equality for ''n'' = 5.
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| == Generalized Ramanujan primes ==
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| Given a constant ''c'' between 0 and 1, the ''n''th ''c''-Ramanujan prime is defined as the
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| 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''
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| 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>''. | |
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| For ''c'' = 1/4 and 3/4, the sequence of ''c''-Ramanujan primes begins
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| :''R''<sub>0.25,''n''</sub> = 2, 3, 5, 13, 17, ... {{OEIS2C|id=A193761}}, | |
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| :''R''<sub>0.75,''n''</sub> = 11, 29, 59, 67, 101, ... {{OEIS2C|id=A193880}}.
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| 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>
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| :''R''<sub>''c'',''n''</sub> ~ ''p''<sub>''n''/(1 − ''c'')</sub> (''n'' → ∞)
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| 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.
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| == Ramanujan prime corollary ==
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| :<math>2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\, ,</math> | |
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| i.e. ''p''<sub>''k''</sub> is the ''k''th prime and the ''n''th Ramanujan prime.
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| 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>).
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| Proof of Corollary:
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| 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
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| ''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''.
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| 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>.
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| An example of this corollary:
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| With ''n'' = 1000, ''R''<sub>''n''</sub> = ''p''<sub>''k''</sub> = 19403, and ''k'' = 2197, therefore ''i'' ≥ 2198 and ''i''−''n'' ≥ 1198.
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| 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.
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| The left side of the Ramanujan Prime Corollary is the {{OEIS2C|id=A168421}}; the right side is the {{OEIS2C|id=A168425}}.
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| The values of <math>\pi(R_n)\,</math> are in the {{OEIS2C|id=A179196}}.
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| The Ramanujan Prime Corollary is due to John Nicholson.
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| ==References==
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| {{Reflist}}
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| {{Prime number classes}}
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| [[Category:Srinivasa Ramanujan]]
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| [[Category:Classes of prime numbers]]
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