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| '''Bertrand's postulate''' (actually a [[theorem]]) states that for any [[integer]] {{nowrap|''n'' > 3,}} there always exists at least one [[prime number]] ''p'' with {{nowrap|''n'' < ''p'' < 2''n'' − 2.}} A weaker but more elegant formulation is: for every {{nowrap|''n'' > 1}} there is always at least one prime ''p'' such that {{nowrap|''n'' < ''p'' < 2''n''.}}
| | Msvcr71.dll is an important file that assists support Windows procedure different components of the system including significant files. Specifically, the file is used to help run corresponding files inside the "Virtual C Runtime Library". These files are important inside accessing any settings that help the different applications and programs in the system. The msvcr71.dll file fulfills many important functions; nevertheless it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer may have a hard time processing plus reading components of the system. Nonetheless, consumers want not panic considering this problem could be solved by following many procedures. And I may show you several strategies regarding Msvcr71.dll.<br><br>However registry is conveniently corrupted plus damaged whenever you may be using your computer. Overtime, without proper maintenance, it is loaded with errors and incorrect or even missing info which can make a program unable to function properly or implement a certain task. And whenever a program may not discover the correct info, it will likely not understand what to do. Then it freezes up! That is the real cause of your trouble.<br><br>System tray icon makes it effortless to launch the program and displays "clean" status or the amount of mistakes inside the last scan. The ability to find plus remove the Invalid class keys and shell extensions is regarded as the primary blessings of the program. That is not routine function for the additional Registry Cleaners. Class keys plus shell extensions which are not functioning could really slow down a computer. RegCure scans to find invalid entries and delete them.<br><br>Chrome allows customizing itself by applying range of themes accessible on the internet. If you had recently applied a theme which no longer functions correctly, it results in Chrome crash on Windows 7. It is recommended to set the original theme.<br><br>The [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] could come because standard with a back up and restore center. This should be an effortless to implement procedure.That signifies that in the event you encounter a issue with your PC following using a registry cleaning we can merely restore your settings.<br><br>Another element is registry. It is regarded as the many important piece inside a Windows XP, Vista running systems. When Windows start, it read connected information from registry plus load into computer RAM. This takes up a big piece of the startup time. After the data is all loaded, computer runs the business programs.<br><br>Maybe you may be asking how come these windows XP error messages appear. Well, for we to be capable to know the fix, you need to first understand where those mistakes come from. There is this software called registry. A registry is software which stores everything on the PC from a regular configuration, setting, info, and logs of activities from installing to UN-installing, saving to deleting, along with a lot more alterations you do in the system pass by it plus gets 'tagged' plus saved because a simple file for recovery purposes. Imagine it as a big recorder, a registrar, of all a records inside your PC.<br><br>Registry products may aid the computer run in a more efficient mode. Registry products ought to be piece of a usual scheduled repair system for your computer. You don't have to wait forever for a computer or the programs to load and run. A small repair usually bring back the speed we lost. |
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| This statement was first conjectured in 1845 by [[Joseph Louis François Bertrand|Joseph Bertrand]] (1822–1900). Bertrand himself verified his statement for all numbers in the interval {{nowrap|[2, 3 × 10<sup>6</sup>].}}
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| His conjecture was completely proved by [[Pafnuty Chebyshev|Chebyshev]] (1821–1894) in 1850 and so the postulate is also called the '''Bertrand–Chebyshev theorem''' or '''Chebyshev's theorem'''. Chebyshev's theorem can also be stated as a relationship with <math>\scriptstyle \pi(x) \,</math>, where <math>\scriptstyle \pi(x) \,</math> is the [[prime counting function]] (number of primes less than or equal to <math>\scriptstyle x \,</math>):
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| :<math>\pi(x) - \pi(\tfrac{x}{2}) \ge 1,\,</math>for all <math>\, x \ge 2. \,</math>
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| In 1919, [[Srinivasa Aaiyangar Ramanujan|Ramanujan]] (1887–1920) used properties of the [[Gamma function]] to give a simpler proof,<ref>{{Cite journal |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 |postscript=<!--None--> }}</ref> from which the concept of [[Ramanujan prime]]s would later arise, and [[Paul Erdős|Erdős]] (1913–1996) in 1932 published a simpler proof using the [[Chebyshev function]] ''ϑ'', defined as:
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| : <math> \vartheta(x) = \sum_{p=2}^{x} \ln (p) </math>
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| where ''p'' ≤ ''x'' runs over primes, and the [[binomial coefficient]]s. See [[proof of Bertrand's postulate]] for the details.
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| == Sylvester's theorem ==
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| Bertrand's postulate was proposed for applications to [[permutation group]]s. [[James Joseph Sylvester|Sylvester]] (1814–1897) generalized it with the statement: the product of ''k'' consecutive integers greater than ''k'' is [[divisible]] by a prime greater than ''k''.
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| == Erdős's theorems ==
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| Erdős proved in 1934 that for any positive integer ''k'', there is a natural number ''N'' such that for all ''n'' > ''N'', there are at least ''k'' primes between ''n'' and 2''n''. An equivalent statement had been proved in 1919 by Ramanujan (see [[Ramanujan prime]]).
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| The [[prime number theorem]] (PNT) implies that the number of primes up to ''x'' is roughly ''x''/log(''x''), so if we replace ''x'' with 2''x'' then we see the number of primes up to 2''x'' is asymptotically twice the number of primes up to ''x'' (the terms log(2''x'') and log(''x'') are asymptotically equivalent). Therefore the number of primes between ''n'' and 2''n'' is roughly ''n''/log(''n'') when ''n'' is large, and so in particular there are many more primes in this interval than are guaranteed by Bertrand's Postulate. So Bertrand's postulate is comparatively weaker than the PNT. But PNT is a deep theorem, while Bertrand's Postulate can be stated more memorably and proved more easily, and also makes precise claims about what happens for small values of ''n''. (In addition, Chebyshev's theorem was proved before the PNT and so has historical interest.)
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| The similar and still unsolved [[Legendre's conjecture]] asks whether for every ''n'' > 1, there is a prime ''p'', such that ''n''<sup>2</sup> < ''p'' < (''n'' + 1)<sup>2</sup>. Again we expect that there will be not just one but many primes between ''n''<sup>2</sup> and (''n'' + 1)<sup>2</sup>, but in this case the PNT doesn't help: the number of primes up to ''x''<sup>2</sup> is asymptotic to ''x''<sup>2</sup>/log(''x''<sup>2</sup>) while the number of primes up to (''x'' + 1)<sup>2</sup> is asymptotic to (''x'' + 1)<sup>2</sup>/log((''x'' + 1)<sup>2</sup>), which is asymptotic to the estimate on primes up to ''x''<sup>2</sup>. So unlike the previous case of ''x'' and 2''x'' we don't get a proof of Legendre's conjecture even for all large ''n''. Error estimates on the PNT are not (indeed, cannot be) sufficient to prove the existence of even one prime in this interval.
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| == Better results ==
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| It follows from the prime number theorem that for any real ε > 0, there exists an ''n''<sub>0</sub> such that there is always a prime between ''n'' and (1 + ε)''n'' for all ''n'' > ''n''<sub>0</sub>: it can be shown, for instance, that | |
| :<math>\lim_{n \to \infty}\frac{\pi((1+\epsilon)n)-\pi(n)}{n/\log n}=\epsilon,</math>
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| which implies that {{nowrap|π((1 + ε)''n'') − π(''n'')}} goes to infinity (and in particular is greater than 1 for sufficiently large ''n'').<ref>G. H. Hardy and E. M. Wright, ''An Introduction to the Theory of Numbers'', 6th ed., Oxford University Press, 2008, p. 494.</ref>
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| Non-asymptotic bounds have also been proved. In 1952, Jitsuro Nagura proved that for ''n'' ≥ 25, there is always a prime between ''n'' and {{nowrap|(1 + 1/5)n}}.<ref>Nagura, J. "On the interval containing at least one prime number." ''Proceedings of the Japan Academy, Series A'' '''28''' (1952), pp. 177–181.[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pja/1195570997&view=body&content-type=pdf_1]</ref>
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| In 1976, [[Lowell Schoenfeld]] showed that for ''n'' ≥ 2010760, there is always a prime between ''n'' and {{nowrap|(1 + 1/16597)''n''}}.<ref>{{cite journal|author=Lowell Schoenfeld|title=Sharper Bounds for the Chebyshev Functions θ(''x'') and ψ(''x''), II|journal=Mathematics of Computation|volume=30|issue=134|pages=337–360|year=April 1976|doi=10.2307/2005976}}</ref> In 1998, [[Pierre Dusart]] improved the result in his doctoral thesis, showing that for ''k'' ≥ 463, {{nowrap|''p''<sub>''k''+1</sub> ≤ (1 + 1/(ln<sup>2</sup>''p<sub>k</sub>''))''p<sub>k</sub>''}}, and in particular for ''x'' ≥ 3275, there exists a prime number between ''x'' and {{nowrap|(1 + 1/(2ln<sup>2</sup>''x''))''x''}}.<ref>{{Citation | last = Dusart | first = Pierre | author-link = Pierre Dusart | url = http://www.unilim.fr/laco/theses/1998/T1998_01.pdf | title = Autour de la fonction qui compte le nombre de nombres premiers | format = PhD thesis | language = french | year = 1998 | format = PDF}}</ref> In 2010 he proved, that for ''x'' ≥ 396738 there is at least one prime between ''x'' and {{nowrap|(1 + 1/(25ln<sup>2</sup>''x''))''x''}}.<ref>{{cite arXiv | last1 = Dusart | first1 = Pierre | authorlink1 = Pierre Dusart | eprint = 1002.0442 | year = 2010 | title = Estimates of Some Functions Over Primes without R.H.}}</ref>
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| Generalizations of Bertrand's Postulate have also been obtained by elementary methods. (In the following, ''n'' runs through the set of positive integers.) In 2006, [[M. El Bachraoui]] proved that there exists a prime between 2''n'' and 3''n''.<ref>M. El Bachraoui, [http://www.m-hikari.com/ijcms-password/ijcms-password13-16-2006/elbachraouiIJCMS13-16-2006.pdf Primes in the Interval (2n, 3n)]</ref> In 2011, [[Andy Loo]] proved that there exists a prime between 3''n'' and 4''n''. Furthermore, he proved that as ''n'' tends to infinity, the number of primes between 3''n'' and 4''n'' also goes to infinity, thereby generalizing Erdős' and Ramanujan's results (see the section on Erdős' theorems above).<ref>{{Citation | url = http://www.m-hikari.com/ijcms-2011/37-40-2011/looIJCMS37-40-2011.pdf | format = PDF | title = On the Primes in the Interval (3''n'', 4''n'') | last = Loo | first = Andy | year = 2011 | journal = International Journal of Contemporary Mathematical Sciences | volume = 6 | issue = 38 | pages = 1871–1882}}</ref> None of these proofs require the use of deep analytic results.
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| ==Consequences==
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| *The sequence of primes, along with 1, is a [[complete sequence]]; any positive integer can be written as a sum of primes (and 1) using each at most once.
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| *The number 1 is the only integer which is a [[harmonic number]].
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| == See also ==
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| * [[Oppermann's conjecture]]
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| ==Notes==
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| <references/>
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| ==Bibliography==
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| * {{cite journal | author=P. Erdős | authorlink=Paul Erdős | title=A Theorem of Sylvester and Schur | journal=[[Journal of the London Mathematical Society]] | volume=9 | issue=4 | pages=282–288 | year=1934 | doi=10.1112/jlms/s1-9.4.282 }}
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| * {{cite journal | doi=10.3792/pja/1195570997 | author=Jitsuro Nagura | title=On the interval containing at least one prime number | journal=Proc. Japan Acad. | volume=28 | issue=4 | pages=177–181 | year=1952}}
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| *{{MathWorld|urlname=BertrandsPostulate|title=Bertrand's Postulate|author=Jonathan Sondow and Eric W. Weisstein}}
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| *Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=BertrandsPostulate ''Bertrand's postulate''] at [[Prime Pages]] glossary.
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| * {{cite journal | author=H. Ricardo | title=Goldbach's Conjecture Implies Bertrand's Postulate | journal=Amer. Math. Monthly | volume=112 | year=2005 | page=492 }}
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| * {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]] | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | page=49 | publisher=Cambridge Univ. Press | location=Cambridge}}
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| {{Prime number classes}}
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| [[Category:Theorems about prime numbers]]
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Msvcr71.dll is an important file that assists support Windows procedure different components of the system including significant files. Specifically, the file is used to help run corresponding files inside the "Virtual C Runtime Library". These files are important inside accessing any settings that help the different applications and programs in the system. The msvcr71.dll file fulfills many important functions; nevertheless it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer may have a hard time processing plus reading components of the system. Nonetheless, consumers want not panic considering this problem could be solved by following many procedures. And I may show you several strategies regarding Msvcr71.dll.
However registry is conveniently corrupted plus damaged whenever you may be using your computer. Overtime, without proper maintenance, it is loaded with errors and incorrect or even missing info which can make a program unable to function properly or implement a certain task. And whenever a program may not discover the correct info, it will likely not understand what to do. Then it freezes up! That is the real cause of your trouble.
System tray icon makes it effortless to launch the program and displays "clean" status or the amount of mistakes inside the last scan. The ability to find plus remove the Invalid class keys and shell extensions is regarded as the primary blessings of the program. That is not routine function for the additional Registry Cleaners. Class keys plus shell extensions which are not functioning could really slow down a computer. RegCure scans to find invalid entries and delete them.
Chrome allows customizing itself by applying range of themes accessible on the internet. If you had recently applied a theme which no longer functions correctly, it results in Chrome crash on Windows 7. It is recommended to set the original theme.
The tuneup utilities could come because standard with a back up and restore center. This should be an effortless to implement procedure.That signifies that in the event you encounter a issue with your PC following using a registry cleaning we can merely restore your settings.
Another element is registry. It is regarded as the many important piece inside a Windows XP, Vista running systems. When Windows start, it read connected information from registry plus load into computer RAM. This takes up a big piece of the startup time. After the data is all loaded, computer runs the business programs.
Maybe you may be asking how come these windows XP error messages appear. Well, for we to be capable to know the fix, you need to first understand where those mistakes come from. There is this software called registry. A registry is software which stores everything on the PC from a regular configuration, setting, info, and logs of activities from installing to UN-installing, saving to deleting, along with a lot more alterations you do in the system pass by it plus gets 'tagged' plus saved because a simple file for recovery purposes. Imagine it as a big recorder, a registrar, of all a records inside your PC.
Registry products may aid the computer run in a more efficient mode. Registry products ought to be piece of a usual scheduled repair system for your computer. You don't have to wait forever for a computer or the programs to load and run. A small repair usually bring back the speed we lost.