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[39] [40] The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. [41] Factorials are used extensively in probability theory, for instance in the Poisson distribution [42] and in the probabilities of random permutations. [43]
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5×4×3×2×1 = 120. By convention, the value of 0! is defined as 1. This classical factorial function appears prominently in many theorems in number theory. The following are a few of these theorems. [1]
The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS). In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0! stands for
To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers 2, 3, 5, and so on, up to the square root of n. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient.
n > 0 is the number of letters in the alphabet (e.g., 26 in English) the falling factorial = (+) denotes the number of strings of length k that don't use any character twice. n! denotes the factorial of n; e = 2.718... is Euler's number; For n = 26, this comes out to 1096259850353149530222034277.
In number theory, a factorion in a given number base is a natural number that equals the sum of the factorials of its digits. [ 1 ] [ 2 ] [ 3 ] The name factorion was coined by the author Clifford A. Pickover .
function factorial is: input: integer n such that n >= 0 output: [n × (n-1) × (n-2) × ... × 1] 1. create new variable called running_total with a value of 1 2. begin loop 1. if n is 0, exit loop 2. set running_total to (running_total × n) 3. decrement n 4. repeat loop 3. return running_total end factorial
function factorial (n is a non-negative integer) if n is 0 then return 1 [by the convention that 0! = 1] else if n is in lookup-table then return lookup-table-value-for-n else let x = factorial(n – 1) times n [recursively invoke factorial with the parameter 1 less than n] store x in lookup-table in the n th slot [remember the result of n! for ...