57 (number)

57 has prime factorization ${\displaystyle 3\cdot 19}$, and is therefore a semiprime.^[1] Its proper divisors are 1, 3, and 19, whose sum is 23, so 57 is a deficient number. Since both prime factors are congruent to 3 modulo 4, ${\displaystyle 57=3\cdot 19}$ is a Blum integer.^[2] It is a Leyland number, because^[3] ${\displaystyle 57=2^{5}+5^{2}}$.

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.^[4]

Although fifty-seven is not prime, it is jokingly known as the Grothendieck prime after a legend in which the mathematician Alexander Grothendieck gave it as an example of a prime number, not realizing it was divisible by three and nineteen.^[5] The same error was made by another famous mathematician, Hermann Weyl, in a published article.^[6]