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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
It follows that arbitrarily large prime numbers can be found as the prime factors of the numbers !, leading to a proof of Euclid's theorem that the number of primes is infinite. [35] When n ! ± 1 {\displaystyle n!\pm 1} is itself prime it is called a factorial prime ; [ 36 ] relatedly, Brocard's problem , also posed by Srinivasa Ramanujan ...
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 .
Several variations on Euclid's proof exist, including the following: The factorial n! of a positive integer n is divisible by every integer from 2 to n, as it is the product of all of them.
The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation!! = ()!! to give !! = (+)!! +.
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.
Since ! is the product of the integers 1 through n, we obtain at least one factor of p in ! for each multiple of p in {,, …,}, of which there are ⌊ ⌋.Each multiple of contributes an additional factor of p, each multiple of contributes yet another factor of p, etc. Adding up the number of these factors gives the infinite sum for (!
Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .
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