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Why is 5040 important?

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 city-state or polis) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a polis.

en.wikipedia.org - 5040 (number) - Wikipedia

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Natural number

5040 is a factorial (7!), a superior highly composite number, abundant number, colossally abundant number and the number of permutations 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 pair.

Philosophy [ edit ]

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 city-state or polis) 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 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, in the introduction to his translation of Laws, wrote, "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 [ edit ]

If σ ( n ) {\displaystyle \sigma (n)} is the divisor function and γ {\displaystyle \gamma } is the Euler–Mascheroni constant, then 5040 is the largest of 27 known numbers (sequence A067698 in the OEIS) for which this inequality holds: σ ( n ) ≥ e γ n log ⁡ log ⁡ n {\displaystyle \sigma (n)\geq e^{\gamma }n\log \log n}

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

lim sup n → ∞ σ ( n ) n log ⁡ log ⁡ n = e γ . {\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 [ edit ]

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).

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How is 1729 a special number?

1729, the Hardy-Ramanujan Number, is the smallest number which can be expressed as the sum of two different cubes in two different ways. 1729 is the sum of the cubes of 10 and 9 - cube of 10 is 1000 and cube of 9 is 729; adding the two numbers results in 1729.

ndtv.com - 1729: The Magic Of Hardy-Ramanujan Number

National Mathematics Day: The Hardy-Ramanujan is probably the easiest to remember among his contributions The man who knew Infinity, Srinivasa Ramanujan knew more than infinity. He contributed theorems and independently compiled 3900 results. However, to inquisitive minds and those dabbling in mathematical science would also know him for the Hardy-Ramanujan number. The Hardy-Ramanujan number is named such after an anecdote of the British mathematician G.H. Hardy who had gone to visit S. Ramanujan in hospital. The anecdote is a part of Ramanujan's biography 'The Man Who Knew Infinity' by Robert Knaigel. Mr. Hardy quipped that he came in a taxi with the number '1729' which seemed a fairly ordinary number. Ramanujan said that it was not. 1729, the Hardy-Ramanujan Number, is the smallest number which can be expressed as the sum of two different cubes in two different ways. 1729 is the sum of the cubes of 10 and 9 - cube of 10 is 1000 and cube of 9 is 729; adding the two numbers results in 1729. 1729 is also the sum of the cubes of 12 and 1- cube of 12 is 1728 and cube of 1 is 1; adding the two results in 1729. While, the Ramanujan number is not his greatest combination, it is certainly a fascinating discovery that is easiest to remember among all of his discoveries. Ramanujan was fascinated with numbers and made striking contributions to a branch of mathematics partitio numerorum, the study of partitions of numbers. Srinivasa Ramanujan's birthday, December 22, was declared as the National mathematics Day in 2012 by the then Prime Minister Dr. Manmohan Singh. Coupled with this and a movie based on his life, he has grabbed the attention of the non-mathematical population as well.

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