Fifty-seven is the sixteenth discrete semiprime and the sixth in the (3×q) family. With 58 it forms the fourth discrete bi-prime pair. 57 has an aliquot sum of 23 and is the first composite member of the 23-aliquot tree. Although 57 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 inNotices of the American Mathematical Society.[1]
There are 57 vertices and 57 hemi-dodecahedral facets in the 57-cell, a 4-dimensional abstract regular polytope.[5] The 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.[6]