5040 (number)

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5040 is a factorial (1 × 2 × 3 × 4 × 5 × 6 × 7 = 5040), and also a highly composite number, a superior highly composite number and a colossally abundant number. It is also the number of permutations of 4 items out of a choice of 10 (10 × 9 × 8 × 7 = 5040). For example, there are 5040 possible 4-digit numbers with all digits different.

5040
List of numbers — Integers ← 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinal	5040 Five thousand [and] forty
Ordinal	5040th Five thousand [and] fortieth
Factorization	$2^4 \cdot 3^2 \cdot 5 \cdot 7$
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
Binary	1001110110000
Duodecimal	2B00
Hexadecimal	13B0

If $σ(n)$ is the divisor function and $γ$ is the Euler-Mascheroni constant, then 5040 is the largest known number for which this inequality holds:

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

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

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

Guy Robin showed in 1984 that the inequality fails for all larger numbers if and only if the Riemann hypothesis is true.

[edit] In other fields

Plato suggested that 5040 was the perfect number of citizens in a state, for various reasons including:

it can be divided by all the numbers 1 to 10 and 12 (and its near neighbour 5031 divides by 11)
it can be divided by 12 twice over.

5040 has been quoted as an important number in some types of numerology. The sum of the radii of the Earth and Moon in miles is 3960 + 1080 = 5040 (in round figures). Twice this figure (the sum of the diameters, or 10,080) is also the number of minutes in a week (7 days × 24 hours × 60 minutes = 10,080).