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In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization.

For a prime factor p of n, the multiplicity of p is the largest exponent a for which p^a divides n. The prime factorization of a positive integer is a list of the integer's prime factors, together with their multiplicity. The fundamental theorem of arithmetic says that every positive integer has a unique prime factorization.

To shorten prime factorization, numbers are often expressed in powers, so

$12 = 2 \times 2 \times 3 = 2^2 \times 3.$

This may not be much of a shortening, but it is in

$288 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^5 \times 3^2.$

For a positive integer n, the number of prime factors of n and the sum of the prime factors of n (not counting multiplicity) are examples of arithmetic functions of n that are additive but not completely additive.

Determining the prime factors of a number is an example of a problem frequently used to ensure cryptographic security in encryption systems; this problem is believed to require super-polynomial time in the number of digits- it is relatively easy to construct a problem that would take longer than the known age of the Universe to calculate on current computers using current algorithms.

Two positive integers are coprime if and only if they have no prime factors in common. The integer 1 is coprime to every positive integer, including itself. This is because it has no prime factors; it is the empty product. It also follows from defining a and b as coprime if gcd(a,b)=1, so that gcd(1,b)=1 for any b>=1. Euclid's algorithm can be used to determine whether two integers are coprime without knowing their prime factors; the algorithm runs in a time that is polynomial in the number of digits involved.

The function $ω(n)$ represents the number of distinct prime factors of n, while $Ω(n)$ represents the total number of prime factors. If $n = \prod_{i=1}^{\omega(n)} p_i^{\alpha_i}$ , then $\Omega(n) = \sum_{i=1}^{\omega(n)} \alpha_i$ .

For example, $24 = 2 3 .3 1$ , so: $ω(24) = 2$ and $Ω(24) = 3 + 1 = 4$ .

ω(n) for n = 1, 2, 3, ... is 0, 1, 1, 1, 1, 2, 1, 1, 1, ... (sequence A001221 in OEIS)

Ω(n) for n = 1, 2, 3, ... is 0, 1, 1, 2, 1, 2, 1, 3, 2, ... (sequence A001222 in OEIS)

[edit] Examples

The prime factors of 6 are 2 and 3 (6 = 2 × 3). Both have multiplicity 1.
5 has only one prime factor: itself (5 is prime). It has multiplicity 1.
100 has two prime factors: 2 and 5 (100 = 2² × 5²). Both have multiplicity 2.
2, 4, 8, 16, etc. each have only one prime factor: 2. (2 is prime, 4 = 2², 8 = 2³, etc.)
1 has no prime factors. (1 is a unit)

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