57 (number)

Fifty-seven is the sixteenth discrete semiprime[1] (specifically, the sixth distinct semiprime of the form ${\displaystyle 3\cdot q}$, where ${\displaystyle q}$ is a higher prime).[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 non-prismatic uniform star polyhedra in 3-space, including four Kepler-Poinsot star polyhedra that are regular.[14]
• 57 vertices and hemi-dodecahedral facets in the 57-cell, a 4-dimensional abstract regular polytope.[15]
• 57 uniform prismatic 5-polytopes in the fifth dimension based on four different finite prismatic families, and inclusive of one special non-Wythoffian figure: the grand antiprism prism.
• 57 uniform prismatic 6-polytopes in the sixth dimension, as prisms of all non-prismatic uniform 5-polytopes.

The split Lie algebra E_7+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 2 of a biographical article on Grothendieck in Notices of the American Mathematical Society.[20]

• The atomic number of lanthanum (La), the first of the lanthanides

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

• "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

• The number of the French department Moselle

• Carnegie Hall is on West 57th Street in New York City

• The model name of a Maybach car
• USS Lake Champlain (CG 57), a Ticonderoga class cruiser in the United States Navy and the third ship to be named Lake Champlain

• The code for international direct dial phone calls to Colombia is 57.
• The number of the laps of the Bahrain Grand Prix.

• List of highways numbered 57