In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

branch of

moonshine theory In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 197 ...

, a supersingular prime is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

that

divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

the order of the

Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 24632059761121331719232931414759 ...

''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. The non-supersingular primes are 37, 43, 53, 61, 67, and any prime number greater than or equal to 73. Supersingular primes are related to the notion of

supersingular elliptic curve In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic ''p'' > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular ar ...

s as follows. For a prime number ''p'', the following are equivalent: # The

modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...

''X''₀⁺(''p'') = ''X''₀(''p'') / ''w''_p, where ''w''_p is the

Fricke involution In mathematics, a Fricke involution is the involution of the modular curve ''X''0(''N'') given by τ → –1/''N''τ. It is named after Robert Fricke. The Fricke involution also acts on other objects associated with the modular curve, suc ...

of ''X''₀(''p''), has

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

zero. # Every supersingular elliptic curve in characteristic ''p'' can be defined over the

prime subfield In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...

F_''p''. # The order of the Monster group is divisible by ''p''. The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that the primes satisfying the first condition are exactly the 15 supersingular primes listed above and shortly thereafter learned of the (then

conjectural In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...

) existence of a sporadic simple group having exactly these primes as prime divisors. This strange coincidence was the beginning of the theory of

monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...

. Three non-supersingular primes occur in the orders of two other sporadic simple groups: 37 and 67 divide the order of the

Lyons group In the area of modern algebra known as group theory, the Lyons group ''Ly'' or Lyons-Sims group ''LyS'' is a sporadic simple group of order : 283756711313767 : = 51765179004000000 : ≈ 5. History ''Ly'' is one of the 26 spor ...

, and 37 and 43 divide the order of the fourth Janko group. It immediately follows that these two are not subquotients of the Monster group (they are two of the six

pariah group In group theory, the term pariah was introduced by Robert Griess in to refer to the six sporadic simple groups which are not subquotients of the monster group In the area of abstract algebra known as group theory, the monster group M (als ...

s). The rest of the sporadic groups (including the other four pariahs, and also the

Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order : 211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group ...

, if that is counted among the sporadics) have orders with only supersingular prime divisors. In fact, other than the

Baby Monster group In the area of modern algebra known as group theory, the baby monster group ''B'' (or, more simply, the baby monster) is a sporadic simple group of order : 241313567211131719233147 : = 4154781481226426191177580544000000 : = 4,1 ...

, they all have orders divisible only by primes less than or equal to 31, although no single sporadic group, other than the Monster itself, has all of them as prime divisors. The supersingular prime 47 also divides the order of the Baby Monster group, and the other three supersingular primes (41, 59, and 71) do not divide the order of any sporadic group other than the Monster itself. All supersingular primes are

Chen prime A prime number ''p'' is called a Chen prime if ''p'' + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2''p'' + 2 therefore satisfies Chen's theorem. The Chen primes are named after Chen Jingru ...

s, but 37, 53, and 67 are also Chen primes, and there are infinitely many Chen primes greater than 73.

References

* * * {{Prime number classes Classes of prime numbers Moonshine theory Sporadic groups fr:Nombre premier super-singulier it:Primi supersingolari