# Prime factor

Language
For the Star Trek: Voyager episode, see Prime Factors (Star Trek: Voyager).

In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's geometric position. He saw a whole number as a line segment, which has a smallest line segment greater than 1 that can divide equally into it.

For a prime factor p of n, the multiplicity of p is the largest exponent a for which pa 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

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

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

For example, $24=2^3 \times 3^1$, so: $\omega(24)=2$ and $\Omega(24)=3+1=4$.

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

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

## See also

• From J.Gabás...

• From THCganja

• From Andy Ciordia

• From christian...

• From “Caveman...

• From woodleywo...

• From epicharmus

• From IRRI Images

• From barnoid

• From tonynetone

• From auntie rain

• From Marjan...

• From Universit...

• From Mark...

• From ohefin

• From The Prime...

• From atelier-ying

• From Robin...

• From Derek K....

• From Chris Devers

• From Chris Devers

• From Chris Devers

• From burtonwoo...

• From twentysix...

• From james_gor...

• From james_gor...

• From andyi

• From Chris Devers

• From Shaojin+AT

• From Eva...

• From Eva...

• From Eva...

• From Eva...

• From Eva...

• From Eva...

• From Eva...

• From Eva...

• From Eva...

• From Madison Guy

• From Tom Purves

• From Christian...

• From lodri

• From aturkus

• From Downing...

• From dkuropatwa

• From IRRI Images

• From IRRI Images

• From IRRI Images

• From IRRI Images

• From IRRI Images

• From IRRI Images

• From PÅL

• From IRRI Images

• From Shayan (USA)

• From Irregular...

• From ttstam

• From s2art

• From detritus

• From Oliver Mayor

• From xJason.Ro...

Images Source: Flickr. Images licensed under the Creative Commons CC-BY-SA
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8695147037506104, "perplexity_flag": "middle"}