Supersingular prime (moonshine theory)

In moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group ${\displaystyle M}$, which is the largest sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31; as well as 41, 47, 59, and 71 (sequence A002267 in the OEIS).

The non-supersingular primes are 37, 43, 53, 61, 67, and all primes greater than or equal to 73.

This usage of "supersingular prime" should not be confused with the related but distinct notion from algebraic number theory. In that context, a prime is called supersingular for a given elliptic curve ${\displaystyle E}$ if reducing ${\displaystyle E}$ modulo ${\displaystyle p}$ yields a supersingular elliptic curve—a property that depends on the choice of curve, and every elliptic curve over ${\displaystyle \mathbb {Q} }$ has infinitely many such primes. By contrast, the fifteen supersingular primes defined here are not relative to any particular curve: they are characterized by a condition on all supersingular elliptic curves in characteristic ${\displaystyle p}$ at once (see below), and happen to coincide with the prime divisors of the order of the Monster group.