5040 (five thousand ndforty) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following 5039 and preceding 5041. It is a

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

(7!), the 8th

superior highly composite number In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to s ...

, the 19th

highly composite number A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(' ...

, an

abundant number In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total ...

, the 8th

colossally abundant number In number theory, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that ...

and the

number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

s of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040). It is also one less than a square, making (7, 71) a

Brown number Brocard's problem is a problem in mathematics that seeks integer values of n such that n!+1 is a perfect square, where n! is the factorial. Only three values of n are known — 4, 5, 7 — and it is not known whether there are any more. ...

pair.

Philosophy

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

mentions in his

dialogue Dialogue (sometimes spelled dialog in American and British English spelling differences, American English) is a written or spoken conversational exchange between two or more people, and a literature, literary and theatrical form that depicts suc ...

Laws Law is a set of rules that are created and are law enforcement, enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a Socia ...

'' that 5040 is a convenient number to use for dividing many things (including both the citizens and the land of a

city-state A city-state is an independent sovereign city which serves as the center of political, economic, and cultural life over its contiguous territory. They have existed in many parts of the world throughout history, including cities such as Rome, ...

or ''

polis Polis (: poleis) means 'city' in Ancient Greek. The ancient word ''polis'' had socio-political connotations not possessed by modern usage. For example, Modern Greek πόλη (polē) is located within a (''khôra''), "country", which is a πατ ...

'') into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a ''polis''. 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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

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

Benjamin Jowett Benjamin Jowett (, modern variant ; 15 April 1817 – 1 October 1893) was an English writer and classical scholar. Additionally, he was an administrative reformer in the University of Oxford, theologian, Anglican cleric, and translator of Plato ...

, in the introduction to his translation of ''Laws'', wrote, "Plato, writing under

Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...

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."

Jean-Pierre Kahane Jean-Pierre Kahane (11 December 1926 – 21 June 2017) was a French mathematician with contributions to harmonic analysis. Career Kahane attended the École normale supérieure and obtained the ''agrégation'' of mathematics in 1949. He then wor ...

has suggested that Plato's use of the number 5040 marks the first appearance of the concept of a

, a number with more divisors than any smaller number..

Number theoretical

\sigma(n)

is the

sum-of-divisors function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...

and

\gamma

is the

Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...

, then 5040 is the largest of 27 known numbers for which this

inequality Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of i ...

holds: :

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

. This is somewhat unusual, since in the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2009 ...

we have: :

\limsup_\frac=e^\gamma.

Guy Robin showed in 1984 that the inequality fails for all larger numbers

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...

is true.

Interesting notes

* 5040 has exactly 60

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

s, counting itself and 1. * 5040 is the largest

(7! = 5040) that is a

. 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 the

least common multiple In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...

of the first 10 multiples of 2 (2, 4, 6, 8, 10, 12, 14, 16, 18 and 20).

References

{{reflist

External links

Mathworld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...

br>article on Plato's numbers
Integers