mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, and more specifically

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, the superfactorial of a positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

n

is the product of the first

n

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

s. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition

The

n

th superfactorial

\mathit(n)

may be defined as:

\begin
\mathit(n) &= 1!\cdot 2!\cdot \cdots n! = \prod_^ i! = n!\cdot\mathit(n-1)\\
&= 1^n \cdot 2^ \cdot \cdots n = \prod_^ i^.\\
\end

Following the usual convention for the

empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...

, the superfactorial of 0 is 1. The

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

of superfactorials, beginning with

\mathit(0)=1

, is:

Properties

Just as the factorials can be continuously interpolated by the

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

, the superfactorials can be continuously interpolated by the Barnes G-function. According to an analogue of

Wilson's theorem In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of ...

on the behavior of factorials

modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...

prime 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 ...

numbers, when

p

is an odd prime number

\mathit(p-1)\equiv(p-1)!!\pmod,

where

!!

is the notation for the

double factorial In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is, n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. Restated ...

. For every integer

k

, the number

\mathit(4k)/(2k)!

is a

square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...

. This may be expressed as stating that, in the formula for

\mathit(4k)

as a product of factorials, omitting one of the factorials (the middle one,

(2k)!

) results in a square product. Additionally, if any

n+1

integers are given, the product of their pairwise differences is always a multiple of

\mathit(n)

, and equals the superfactorial when the given numbers are consecutive.

References

External links

*{{MathWorld, id=Superfactorial, title=Superfactorial, mode=cs2 Integer sequences Factorial and binomial topics