|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
integer factorization
|
(Definition)
|
|
|
Given an integer  , its integer factorization (or prime factorization) consists of the primes  which multiplied together give  as a result. To put it algebraically,
with each  distinct, all  but not necessarily distinct, and  being the value of the number of distinct prime factors function.
For example, the factorization of 32851 is
 . Because of the commutative property of multiplication, it does not matter in what order the prime factors 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  one could take the factorization of  and randomly give negative signs to one (or any odd number) of the prime factors. Alternatively, the factorization can be given as
 (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 algorithms to choose from, with trial division being the simplest to implement.
|
"integer factorization" is owned by PrimeFan.
|
|
(view preamble)
| Other names: |
prime factorization |
This object's parent.
|
|
Cross-references: trial division, algorithms, term, Mathematica, odd number, negative, negative number, fundamental theorem of arithmetic, positive, exponents, group, ascending order, prime factors, order, multiplication, property, commutative, number of distinct prime factors function, primes, integer
There are 31 references to this entry.
This is version 4 of integer factorization, born on 2007-02-02, modified 2007-07-11.
Object id is 8857, canonical name is IntegerFactorization.
Accessed 2414 times total.
Classification:
| AMS MSC: | 11A41 (Number theory :: Elementary number theory :: Primes) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
