5040 (five thousand [and] forty) is thenatural number following 5039 and preceding 5041.

It is afactorial (7!), the 8thsuperior highly composite number,^[1] the 19thhighly composite number,^[2] anabundant number, the 8thcolossally abundant number^[3] and thenumber ofpermutations of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040). It is also one less than a square, making (7, 71) aBrown number pair.

Plato mentions in hisdialogueLaws that 5040 is a convenient number to use fordividing many things (including both the citizens and the land of acity-state orpolis) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up apolis.^[4] He remarks that this number can be divided by all the(natural) numbers from1 to12 with the single exception of11 (however, it is not the smallest number to have this property;2520 is). He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which isdivisible 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 thatBenjamin Jowett, in the introduction to his translation ofLaws, wrote, "Plato, writing underPythagorean 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."^[5]

Jean-Pierre Kahane has suggested that Plato's use of the number 5040 marks the first appearance of the concept of ahighly composite number, a number with more divisors than any smaller number.^[6]

If${\displaystyle \sigma (n)}$ is thesum-of-divisors function and${\displaystyle \gamma }$ is theEuler–Mascheroni constant, then 5040 is the largest of 27 known numbers (sequenceA067698 in theOEIS) for which thisinequality holds:

${\displaystyle \sigma (n)\ge e^{\gamma }n\log \log n}$.

This is somewhat unusual, since in thelimit we have:

${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}\frac{\sigma (n)}{n\log \log n}=e^{\gamma }.}$

Guy Robin showed in 1984 that the inequality fails for all larger numbersif and only if theRiemann hypothesis is true.^{[citation needed]}

• 5040 has exactly 60divisors, counting itself and1.
• 5040 is the largestfactorial (7! = 5040) that is ahighly composite number. All factorials smaller than 8! =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 theleast common multiple of the first 10 multiples of 2 (2, 4, 6, 8, 10, 12, 14, 16, 18 and 20).

• Mathworldarticle on Plato's numbers

• Integers

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