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The factorial function of a positive integer is defined by the product of all positive integers not greater than [1] this may be written more concisely in product notation as [1] if this product formula is changed to keep all but the last term, it would define a product of the same form, for a smaller factorial

This leads to a recurrence relation, according to which each value of the. Factorial for any positive integer n, the product of the integers less than or equal to n is a unary operation called factorial In the context of complex numbers, the gamma function is a unary operation extension of factorial. The rising factorial is also integral to the definition of the hypergeometric function The hypergeometric function is defined for by the power series provided that Note, however, that the hypergeometric function literature typically uses the notation for rising factorials.

In combinatorics, the factorial number system (also known as factoradic), is a mixed radix numeral system adapted to numbering permutations It is also called factorial base, although factorials do not function as base, but as place value of digits By converting a number less than n To factorial representation, one obtains a sequence of n digits that can be converted to a permutation of n. Relation to the factorial because the double factorial only involves about half the factors of the ordinary factorial, its value is not substantially larger than the square root of the factorial n!, and it is much smaller than the iterated factorial (n!)!. Why are trailing fractional zeros important

In the classical untyped lambda calculus, every function has a fixed point

A particular implementation of is haskell curry 's paradoxical combinator y, given by [2] 131 [note 1][note 2] (here using the standard notations and conventions of lambda calculus Y is a function that takes one argument f and returns the entire expression following the first period The expression denotes a function. 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 modular arithmetic), the factorial satisfies exactly when n is a prime number

In other words, any integer n > 1 is a prime number if, and only if, (n − 1) Let be a natural number For a base , we define the sum of the factorials of the digits[5][6] of , , to be the following Sfd b ⁡ ( n ) = ∑ i = 0 k − 1 d i Factorial formula finally, though computationally unsuitable, there is the compact form, often used in proofs and derivations, which makes repeated use of the familiar factorial function Denotes the factorial of n.