| ||||
---|---|---|---|---|
Cardinal | fifty-seven | |||
Ordinal | 57th (fifty-seventh) | |||
Factorization | 3 × 19 | |||
Divisors | 1, 3, 19, 57 | |||
Greek numeral | ΝΖ´ | |||
Roman numeral | LVII | |||
Binary | 1110012 | |||
Ternary | 20103 | |||
Senary | 1336 | |||
Octal | 718 | |||
Duodecimal | 4912 | |||
Hexadecimal | 3916 |
57 (fifty-seven) is the natural number following 56 and preceding 58.
Fifty-seven is the sixteenth discrete semiprime [1] (specifically, the sixth semiprime of the form , where is a prime strictly larger than 3). [2] It also forms the fourth discrete semiprime pair with 58.
57 is the third Blum integer since its two prime factors ( 3 and 19) are both Gaussian primes. [3] 57 has an aliquot sum of 23, which makes it the tenth number to contain a prime aliquot sum. [4] This also makes 57 the first composite member of the 23-aliquot tree (..., 57, 23, 1, 0). The only other numbers to generate an aliquot sum of 57 are 99, 159, 343, 559, and 703; [5] where 343 is the cube of 7, [6] and 703 the sum of the first thirty-seven nonzero integers. [7] Fifty seven is also a repdigit in base-7 (111). [8]
57 is the fifth Leyland number, as it can be written in the form: [9]
57 is the number of compositions of 10 into distinct parts. [10]
57 is the seventh fine number, equivalently the number of ordered rooted trees with seven nodes having root of even degree. [11]
57 is also the number of nodes in a regular octagon when all of its diagonals are drawn, [12] and the first non-trivial icosagonal (20-gonal) number. [13]
In geometry, there are:
The split Lie algebra E7+1/2 has a 57-dimensional Heisenberg algebra as its nilradical, and the smallest possible homogeneous space for E8 is also 57-dimensional. [16]
57 lies between prime numbers 53 and 61, which are the only two prime numbers less than 71 that do not divide the order of any sporadic group, inclusive of the six pariahs. 71, the twentieth prime number, is the largest supersingular prime that divides the largest of these groups [17] while 57, on the other hand, is the fortieth composite number whose sum of divisors σ(57) is 80 and averages 20. [18] [19]
Although fifty-seven is not prime, it is jokingly known as the Grothendieck prime after a story in which mathematician Alexander Grothendieck supposedly gave it as an example of a particular prime number. This story is repeated in Part II of a biographical article on Grothendieck in Notices of the American Mathematical Society. [20] However, its veracity is questionable, and it may be a confounded misattribution of the same blunder committed to writing by Hermann Weyl. [21]
| ||||
---|---|---|---|---|
Cardinal | fifty-seven | |||
Ordinal | 57th (fifty-seventh) | |||
Factorization | 3 × 19 | |||
Divisors | 1, 3, 19, 57 | |||
Greek numeral | ΝΖ´ | |||
Roman numeral | LVII | |||
Binary | 1110012 | |||
Ternary | 20103 | |||
Senary | 1336 | |||
Octal | 718 | |||
Duodecimal | 4912 | |||
Hexadecimal | 3916 |
57 (fifty-seven) is the natural number following 56 and preceding 58.
Fifty-seven is the sixteenth discrete semiprime [1] (specifically, the sixth semiprime of the form , where is a prime strictly larger than 3). [2] It also forms the fourth discrete semiprime pair with 58.
57 is the third Blum integer since its two prime factors ( 3 and 19) are both Gaussian primes. [3] 57 has an aliquot sum of 23, which makes it the tenth number to contain a prime aliquot sum. [4] This also makes 57 the first composite member of the 23-aliquot tree (..., 57, 23, 1, 0). The only other numbers to generate an aliquot sum of 57 are 99, 159, 343, 559, and 703; [5] where 343 is the cube of 7, [6] and 703 the sum of the first thirty-seven nonzero integers. [7] Fifty seven is also a repdigit in base-7 (111). [8]
57 is the fifth Leyland number, as it can be written in the form: [9]
57 is the number of compositions of 10 into distinct parts. [10]
57 is the seventh fine number, equivalently the number of ordered rooted trees with seven nodes having root of even degree. [11]
57 is also the number of nodes in a regular octagon when all of its diagonals are drawn, [12] and the first non-trivial icosagonal (20-gonal) number. [13]
In geometry, there are:
The split Lie algebra E7+1/2 has a 57-dimensional Heisenberg algebra as its nilradical, and the smallest possible homogeneous space for E8 is also 57-dimensional. [16]
57 lies between prime numbers 53 and 61, which are the only two prime numbers less than 71 that do not divide the order of any sporadic group, inclusive of the six pariahs. 71, the twentieth prime number, is the largest supersingular prime that divides the largest of these groups [17] while 57, on the other hand, is the fortieth composite number whose sum of divisors σ(57) is 80 and averages 20. [18] [19]
Although fifty-seven is not prime, it is jokingly known as the Grothendieck prime after a story in which mathematician Alexander Grothendieck supposedly gave it as an example of a particular prime number. This story is repeated in Part II of a biographical article on Grothendieck in Notices of the American Mathematical Society. [20] However, its veracity is questionable, and it may be a confounded misattribution of the same blunder committed to writing by Hermann Weyl. [21]