# 5040 (number)

 ← 5039 5040 5041 →
Cardinal five thousand forty
Ordinal 5040th
(five thousand fortieth)
Factorization 24× 32× 5 × 7
Divisors 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
Roman numeral VXL
Binary 10011101100002
Ternary 202202003
Quaternary 10323004
Quinary 1301305
Senary 352006
Octal 116608
Duodecimal 2B0012
Vigesimal CC020
Base 36 3W036

5040 is a factorial (7!), a superior highly composite number, a colossally abundant number, and the number of permutations of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040).

## Philosophy

Plato mentions in his Laws that 5040 is a convenient number to use for 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 from 1 to 12 with the single exception of 11 (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 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 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."[1]

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.[2]

## Number theoretical

If ${\displaystyle \sigma (n)}$ is the divisor function and ${\displaystyle \gamma }$ is the Euler–Mascheroni constant, then 5040 is the largest of the known numbers (sequence A067698 in the OEIS) for which this inequality holds:

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

This is somewhat unusual, since in the limit we have:

${\displaystyle \limsup _{n\rightarrow \infty }{\frac {\sigma (n)}{n\ \log \log n}}=e^{\gamma }.}$

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 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 radii of both the Earth and Moon (in miles) is 3960 + 1080 = 5040.[3] Incidentally, the sum of their diameters 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 − 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.
• 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

1. ^ Laws, by Plato at Project Gutenberg; retrieved 7 July 2009
2. ^ Kahane, Jean-Pierre (February 2015), "Bernoulli convolutions and self-similar measures after Erdős: A personal hors d'oeuvre", Bulletin of the American Mathematical Society, 62 (2): 136–140.
3. ^ City of Revelation: On the Proportions and Symbolic Numbers of the Cosmic Temple, by John Michell (ISBN 0-345-23607-6), p. 61.