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In combinatorics, the factorial number system (also known as factoradic), is a mixed radix numeral system adapted to numbering permutations ± k is divisible by k for 2 ≤ k ≤ n. 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 Itself, also, each number of form n To factorial representation, one obtains a sequence of n digits that can be converted to a permutation of n.

Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials

Daniel bernoulli and leonhard euler interpolated the factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function. Multiplicative partitions of factorials are expressions of values of the factorial function as products of powers of prime numbers They have been studied by paul erdős and others [1][2][3] the factorial of a positive integer is a product of decreasing integer factors, which can in turn be factored into prime numbers.

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 These are counted by the double factorial 15 = (6 − 1)‼

In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n

Falling factorials appear in multiple differentiation of simple power functions ( d d x ) n x a = ( a ) n ⋅ x a − n The hypergeometric function is defined for by the power series provided that. The stirling numbers of the first kind are the coefficients in the expansion of the falling factorial into powers of the variable

For example, , leading to the values , , and The unsigned stirling numbers may also be defined algebraically as the coefficients of the rising factorial The notations used on this page for stirling numbers are not universal, and may conflict with notations in. 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 modular arithmetic), the factorial satisfies exactly when n is a prime number.

No other factorial primes are known as of june 2025 − 1 are composite, there must be at least 2 n + 1 consecutive composite numbers around n!, since besides n

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