# 57 (number)

 ← 56 57 58 →
Cardinalfifty-seven
Ordinal57th
(fifty-seventh)
Factorization3 × 19
Divisors1, 3, 19, 57
Greek numeralΝΖ´
Roman numeralLVII
Binary1110012
Ternary20103
Senary1336
Octal718
Duodecimal4912

57 (fifty-seven) is the natural number following 56 and preceding 58.

## In mathematics

Fifty-seven is the sixteenth discrete semiprime[1] (specifically, the sixth semiprime of the form ${\displaystyle 3\cdot q}$, where ${\displaystyle q}$ 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]

${\displaystyle 5^{2}+2^{5}=57}$

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:

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 legend according to which the mathematician Alexander Grothendieck supposedly gave it as an example of a particular prime number.[20] The joke is that he is famous for working abstractly, without concrete examples. However, while the veracity of this legend about Grothendieck is unclear, it is known that this very error was committed by another famous mathematician Hermann Weyl in a published article.[21]

## In fiction and media

### In films

• Passenger 57, a film starring Wesley Snipes
• In the movie Contagion, Vaccine #57 successfully protects the lab monkey from infection.
• The Terminal (2004) starring Tom Hanks. There are 57 members of the jazz band that Viktor Navorski carries a picture of with him.
• in the movie Eraser (1996), the weapons trade operation took place at the Baltimore Docks, Pier 57.

### In games

• In the game Hollow Knight, a character named Zote the Mighty has 57 precepts, all of which offer rather humorous, lackluster, or completely bad advice to the player.

## In food

• Heinz 57, a brand of sauce, and the number of varieties of foods claimed to be produced by the H.J. Heinz Company

## In music

• "Incident on 57th Street", a song by Bruce Springsteen and the E Street Band, from their 1973 album, The Wild, the Innocent and the E Street Shuffle
• "57 Channels (And Nothin' On)", a song by Bruce Springsteen, from his 1992 album Human Touch
• "57", the name of a song by Biffy Clyro on their 2002 debut album, Blackened Sky
• Shure SM57, considered the workhorse of recording microphones

## In organizations

• The number of the French department Moselle

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A001358 (Semiprimes (or biprimes): products of two primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^ Sloane, N. J. A. (ed.). "Sequence A001748 (Semiprimes of the form a(n) equal to 3 x prime(n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. ^ Sloane, N. J. A. (ed.). "Sequence A016105 (Blum integers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. ^
5. ^
6. ^ Sloane, N. J. A. (ed.). "Sequence A000578 (The cubes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. ^ Sloane, N. J. A. (ed.). "Sequence A048332 (Numbers that are repdigits in base 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
9. ^ Sloane, N. J. A. (ed.). "Sequence A076980 (Leyland numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
10. ^
11. ^ Sloane, N. J. A. (ed.). "Sequence A000957 (Fine's sequence (or Fine numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-06-01.
12. ^
13. ^ Sloane, N. J. A. (ed.). "Sequence A051872 (20-gonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
14. ^ Skilling, J. (1975). "The complete set of uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. 278 (1278): 111–135. Bibcode:1975RSPTA.278..111S. doi:10.1098/rsta.1975.0022. ISSN 0080-4614. JSTOR 74475. MR 0365333. S2CID 122634260.
15. ^ Coxeter, H. S. M. (1982), "Ten toroids and fifty-seven hemidodecahedra", Geometriae Dedicata, 13 (1): 87–99, doi:10.1007/BF00149428, MR 0679218, S2CID 120672023
16. ^ Vogan, David (2007), "The character table for E8" (PDF), Notices of the American Mathematical Society, 54 (9): 1122–1134, MR 2349532
17. ^
18. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
19. ^
20. ^ Jackson, Allyn (2004b). "Comme Appelé du Néant—As if Summoned from the Void: The Life of Alexandre Grothendieck" (PDF). Notices of the American Mathematical Society. 51 (10). Providence, RI: American Mathematical Society: 1196, 1197. MR 2104915. Zbl 1168.01339.
21. ^ Weyl, Hermann (1951). "A Half-Century of Mathematics". American Mathematical Monthly. 58 (5). Washington, D.C.: Mathematical Association of America: 532. doi:10.1080/00029890.1951.11999734. JSTOR 2306319. S2CID 126101329.
22. ^ The NGC / IC Project - Home of the Historically Corrected New General Catalogue (HCNGC) since 1993