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Integer Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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, this says that for even , the double factorial is n!! = \prod_^\frac (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,, while for odd it is n!! = \prod_^\frac (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,. For example, . The zero double factorial as an empty product. The sequence of double factorials for even = starts as The sequence of double factorials for odd = starts as The term odd factorial is sometimes used for the double factorial of an odd number. The term semifactorial is also used by Donald Knuth, Knuth as a synonym of double factorial. History and usage In a 1902 paper, the physicist Arthur Schuster wrote: states that the double factorial was originally introduced in order to simplify the expression of certain List of integrals of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |