integer factorization

Given an integer n, its integer factorization (or prime factorization) consists of the primes pi which multiplied together give n as a result. To put it algebraically,


with each pi distinct, all ai>0 but not necessarily distinct, and ω(n) being the value of the number of distinct prime factors function. Theoretically, an integer is a product of all the prime numbersMathworldPlanetmath,


with many ai=0.

For example, the factorization of 32851 is 7×13×19×19, more usually expressed as 7×13×192. Because of the commutative property of multiplication, it does not matter in what order the prime factorsMathworldPlanetmath are stated in, but it is customary to give them in ascending order (, and to group them together by the use of exponents.

The factorization of a positive integer is unique (this is the fundamental theorem of arithmetic). For a negative number n<0 one could take the factorization of |n| and randomly give negative signs to one (or any odd numberMathworldPlanetmathPlanetmath) of the prime factors. Alternatively, the factorization can be given as -1p1a1 (this is what Mathematica opts for).

The term “factorization” is often used to refer to the actual process of determining the prime factors. There are several algorithmsMathworldPlanetmath to choose from, with trial divisionMathworldPlanetmath being the simplest to implement.

Title integer factorization
Canonical name IntegerFactorization
Date of creation 2013-03-22 16:39:09
Last modified on 2013-03-22 16:39:09
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 8
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A41
Synonym prime factorization