View source for 5040 (number)

{{Infobox number
| number =  5040
| 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
}}
'''5040''' is a [[factorial]] (7!), 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).

==Philosophy==
[[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]] (however, it is not the smallest number to have this property; [[2520 (number)|2520]] is).  He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number [[5038 (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>

[[Jean-Pierre Kahane]] has suggested that Plato's use of the number 5040 marks the first appearance of the concept of a [[highly composite number]], a number with more divisors than any smaller number.<ref>{{citation|first=Jean-Pierre|last=Kahane|authorlink=Jean-Pierre Kahane|title=Bernoulli convolutions and self-similar measures after Erdős: A personal hors d'øeuvre|journal=Bulletin of the American Mathematical Society|date=February 2015|volume=62|issue=2|pages=136–140}}.</ref>

==Number theoretical==
If <math>\sigma(n)</math> is the [[divisor function]] and <math>\gamma</math> is the [[Euler–Mascheroni constant]], then 5040 is the largest of the known numbers {{OEIS| A067698}} for which this [[Inequality (mathematics)|inequality]] holds:
:<math>\sigma(n) \geq e^\gamma n\log \log n </math>.
This is somewhat unusual, since in the [[limit (mathematics)|limit]] we have:
:<math>\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\ \log \log n}=e^\gamma.</math>
Guy Robin showed in 1984 that the inequality fails for all larger numbers [[if and only if]] the [[Riemann hypothesis]] is true.

==Interesting notes==
* 5040 has exactly 60 divisors, counting itself and 1.
* 5040 is the largest [[factorial]] (7! = 5040) that is also a [[highly composite number]]. All factorials smaller than 8! = [[40320 (number)|40320]] are highly composite.
* 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).
* 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).
* 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&nbsp;−&nbsp;[[π|{{pi}}]])/{{pi}}, when using 22/7 as the value of&nbsp;{{pi}}. This means that the sizes of the earth and the moon are related by a simple function of&nbsp;{{pi}}.
* 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.

==Notes==
<references/>

==External links==
* [[Mathworld]] [http://mathworld.wolfram.com/PlatosNumbers.html article on Plato's numbers]

[[Category:Integers]]
[[Category:Plato]]