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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''.}}
| | BMI is short for "body mass index". It's calculated from a person's weight inside kilograms and height in meters. The easiest method to determine BMI is to use an online calculator where you plug in your weight plus height. A usual BMI is between 18.5 plus 24.9, whilst 25 to 29.9 is considered to be obese. Once you hit a BMI of 30, you're overweight. On the additional hand, these values are less exact in athletic persons that have more lean body mass plus folks who have big frames.<br><br>To calculate the waist-to-hip ratio (to keep an acronym it's called the WTHR), measure both the waist plus hips, plus divide the waist size by the cool size. And yes, this indicator realizes the difference between guys and women, unlike the BMI. For females, the ratio must be no over 0.8. In additional words, the waist ought to be smaller than the hips. And for men, it ought to be 1.0 or less. That signifies a man's waist ought to be the same measurement because his hips or small. The beer-and-pizza stomach has to go!<br><br>He actually had kind 1.5 diabetes or Latent autoimmune diabetes of adults (LADA). Here is what Wikipedia claims about LADA. Officially it may nevertheless be mentioned to be sort 1 diabetes. This really is a smaller article of mine, but you are able to watch the 2 videos below plus you are able to moreover buy Dr. Cousen's book on treating diabetes.<br><br>In addition, men generally have more muscle than ladies plus women have more fat inside their body than men. For this reason, females must have their fat plus height ratio calculated by a [http://safedietplansforwomen.com/bmi-calculator bmi calculator women] for ladies.<br><br>This is the case with a person i understand. She's 63, 5'9" and surprisingly close to 300 lbs. While she could walk, inside a limited manner, her efforts to lose weight are just not working. She has additional problems too like CHF and osteoporosis. Her target weight is around 180, so thats over 100 lbs she must loose. I devised this plan for her!<br><br>In order to determine ideal weight for kids plus teenagers (between age group 2-20 years), there is a certain reference tool called the BMI percentile chart. In case, if the percent value falls at 80, it means the kid is having more fat than 80 % of the kids of the same gender and age group. According to the chart, a child's body mass index falling in 95 percentile or above indicates he is obese, while those with a BMI above 85 percentile have a risk of becoming obese. On the different hand, when a kid's BMI percentile is 5 or lower, he/she is underweight.<br><br>In terms of its dodgy mathematics, the BMI is borderline junk science. However inside this regard, BMI is less egregious than Global Warming 'studies', in which nearly all of the big-name 'researchers' cherry-pick, hide, or fabricate information. Click on the link for my lengthy hub found on the topic. |
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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
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| :<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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BMI is short for "body mass index". It's calculated from a person's weight inside kilograms and height in meters. The easiest method to determine BMI is to use an online calculator where you plug in your weight plus height. A usual BMI is between 18.5 plus 24.9, whilst 25 to 29.9 is considered to be obese. Once you hit a BMI of 30, you're overweight. On the additional hand, these values are less exact in athletic persons that have more lean body mass plus folks who have big frames.
To calculate the waist-to-hip ratio (to keep an acronym it's called the WTHR), measure both the waist plus hips, plus divide the waist size by the cool size. And yes, this indicator realizes the difference between guys and women, unlike the BMI. For females, the ratio must be no over 0.8. In additional words, the waist ought to be smaller than the hips. And for men, it ought to be 1.0 or less. That signifies a man's waist ought to be the same measurement because his hips or small. The beer-and-pizza stomach has to go!
He actually had kind 1.5 diabetes or Latent autoimmune diabetes of adults (LADA). Here is what Wikipedia claims about LADA. Officially it may nevertheless be mentioned to be sort 1 diabetes. This really is a smaller article of mine, but you are able to watch the 2 videos below plus you are able to moreover buy Dr. Cousen's book on treating diabetes.
In addition, men generally have more muscle than ladies plus women have more fat inside their body than men. For this reason, females must have their fat plus height ratio calculated by a bmi calculator women for ladies.
This is the case with a person i understand. She's 63, 5'9" and surprisingly close to 300 lbs. While she could walk, inside a limited manner, her efforts to lose weight are just not working. She has additional problems too like CHF and osteoporosis. Her target weight is around 180, so thats over 100 lbs she must loose. I devised this plan for her!
In order to determine ideal weight for kids plus teenagers (between age group 2-20 years), there is a certain reference tool called the BMI percentile chart. In case, if the percent value falls at 80, it means the kid is having more fat than 80 % of the kids of the same gender and age group. According to the chart, a child's body mass index falling in 95 percentile or above indicates he is obese, while those with a BMI above 85 percentile have a risk of becoming obese. On the different hand, when a kid's BMI percentile is 5 or lower, he/she is underweight.
In terms of its dodgy mathematics, the BMI is borderline junk science. However inside this regard, BMI is less egregious than Global Warming 'studies', in which nearly all of the big-name 'researchers' cherry-pick, hide, or fabricate information. Click on the link for my lengthy hub found on the topic.