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| {{Infobox number
| | School Principal Guillermo Kania from Harrow, really likes relaxing, hair loss and boomerangs. In the last month or two has visited to locations such as Wolong.<br><br>my weblog: [http://scarceguru9561.blox.pl/2015/03/Hair-Loss-Protocol-Presents-Revolutionary.html hair loss cure 2013] |
| | number = 5040
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| | divisor = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040
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| }}
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| '''5040''' is a [[factorial]] (7!), a [[highly composite number]], a [[superior highly composite number]], a [[colossally abundant number]], and the [[number]] of [[permutation]]s of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040).
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| ==Philosophy==
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| [[Plato]] mentions in his [[Laws (dialogue)|Laws]] that 5040 is a convenient number to use for [[division (mathematics)|dividing]] many things (including both the citizens and the land of a state) into lesser parts. He remarks that this number can be divided by all the [[natural numbers|(natural) numbers]] from [[1 (number)|1]] to [[12 (number)|12]] with the single exception of [[11 (number)|11]]. He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which is [[divisible]] by 11. Plato also took notice of the fact that 5040 can be divided by 12 twice over. Indeed, Plato's repeated insistence on the use of 5040 for various state purposes is so evident that it is written, "Plato, writing under [[Pythagoreanism|Pythagorean]] influences, seems really to have supposed that the well-being of the city depended almost as much on the number 5040 as on [[justice]] and [[moderation]]."<ref>[http://www.gutenberg.org/files/1750/1750-h/1750-h.htm Laws], by Plato at Project Gutenberg; retrieved 7 July 2009</ref>
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| ==Number Theory==
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| If <math>\sigma(n)</math> is the [[divisor function]] and <math>\gamma</math> is the [[Euler-Mascheroni constant]], then 5040 is the largest known number for which this [[Inequality (mathematics)|inequality]] holds:
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| :<math>\sigma(n) \geq e^\gamma n\log \log n </math>.
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| This is somewhat unusual, since in the [[limit (mathematics)|limit]] we have:
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| :<math>\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\ \log \log n}=e^\gamma.</math>
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| Guy Robin showed in 1984 that the inequality fails for all larger numbers [[if and only if]] the [[Riemann hypothesis]] is true.
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| ==Other==
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| ▪ 5040 is considered an important number in some systems of [[numerology]], not only because of the Plato connection, but because using round figures, the sum of the [[radius|radii]] of both the [[Earth]] and [[Moon]] (in miles) is 3960 + 1080 = 5040.<ref>''City of Revelation: On the Proportions and Symbolic Numbers of the Cosmic Temple'', by [[John Michell (writer)|John Michell]] (ISBN 0-345-23607-6), p. 61.</ref> Incidentally, the sum of their [[diameter]]s is also the number of minutes in a week (7 days × 24 hours × 60 minutes = 10,080).
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| ▪ The [[ratio]] of the [[radius]] of the moon and the radius of the earth is 1080/3960, which simplifies to 3/11. This ratio can also be expressed as (4 - [[π]])/π, when using 22/7 as the value of π. This means that the sizes of the earth and the moon are related by a simple function of π.
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| ▪ Given that the radius of the moon is 3/11 that of the earth, the sum of their radii can be broken into 3/14 (for the radius of the moon) and 11/14 (for the radius of the earth). Further, the sum of their radii in miles is 5040, which when divided by 14 is 360 (the number of degrees in a circle). This would not happen for another pair of objects with radii in the same ratio - it only happens when the sum of their radii is 5040.
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| ▪ 5040 has exactly 60 divisors, counting itself and 1.
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| ▪ In a [[vigesimal]] system, 5040 is represented as 12 groups of 20 and 12 groups of 20-squared (12 • 20 = 240, 12 • 20<sup>2</sup> = 4800, and 240 + 4800 = 5040).
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| ▪ 5040 is the sum of 42 consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 +163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229).
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| ==Notes==
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| <references/>
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| ==External links==
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| * [[Mathworld]] [http://mathworld.wolfram.com/PlatosNumbers.html article on Plato's numbers]
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| [[Category:Integers|59e03 5040]]
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| [[Category:Plato]]
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School Principal Guillermo Kania from Harrow, really likes relaxing, hair loss and boomerangs. In the last month or two has visited to locations such as Wolong.
my weblog: hair loss cure 2013