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{{For|the ''[[Star Trek: Voyager]]'' episode|Prime Factors (Star Trek: Voyager)}}
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{{refimprove|date=July 2013}}
 
In [[number theory]], the '''prime factors''' of a positive  [[integer]] are the [[prime number]]s that divide that integer exactly. The [[integer factorization|prime factorization]] of a positive integer is a list of the integer's prime factors, together with their [[Multiplicity (mathematics)|multiplicities]]; the process of determining these factors is called [[integer factorization]]. The [[fundamental theorem of arithmetic]] says that every positive integer has a single unique prime factorization.
 
To shorten prime factorizations, factors are often expressed in powers (multiplicities). For example,
:<math> 360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5,</math>
in which the factors 2, 3 and 5 have multiplicities of 3, 2 and 1, respectively.
 
For a prime factor ''p'' of ''n'', the multiplicity of ''p'' is the largest [[Exponentiation|exponent]] ''a'' for which ''p<sup>a</sup>'' divides ''n'' exactly.
 
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 function]]s of ''n'' that are [[additive function|additive]] but not completely additive.
 
==Perfect squares==
[[Square number|Perfect square numbers]] can be recognized by the fact that all of their prime factors have even multiplicities. For example, the number 144 (the square of 12) has the prime factors
:<math> 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2.</math>
These can be rearranged to make the pattern more visible:
:<math> 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = (2 \times 2 \times 3) \times (2 \times 2 \times 3) = (2 \times 2 \times 3)^2 = (12)^2.</math>
Because every prime factor appears an even number of times, the original number can be expressed as the square of some smaller number. In the same way, [[Cube number|perfect cube numbers]] will have prime factors whose multiplicities are multiples of three, and so on.
 
==Coprime integers==
Positive integers with no prime factors in common are said to be [[coprime]]. Two integers ''a'' and ''b'' can also be defined as coprime if their [[greatest common divisor]] gcd(''a'',&nbsp;''b'')&nbsp;=&nbsp;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_complexity#Polynomial_time|time that is polynomial]] in the number of digits involved.
 
The integer 1 is coprime to every positive integer, including itself. This is because it has no prime factors; it is the [[empty product]]. This implies that gcd(1,&nbsp;''b'')&nbsp;=&nbsp;1 for any ''b''&nbsp;≥&nbsp;1.
 
==Cryptographic Applications==
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 [[Time_complexity#Superpolynomial_time|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.
 
== {{anchor|Omega function}} Omega functions ==
The function {{math|ω(''n'')}} ''("omega")'' represents the number of ''distinct'' prime factors of ''n'', while the function {{math|Ω(''n'')}} ''("big omega")'' represents the ''total'' number of prime factors of {{math|''n''}}.
If
:<math>n = \prod_{i=1}^{\omega(n)} p_i^{\alpha_i}</math>,
then
:<math>\Omega(n) = \sum_{i=1}^{\omega(n)} \alpha_i</math>.
 
For example, {{math|1=24 = 2<sup>3</sup> × 3<sup>1</sup>}}, so {{math|1=ω(24) = 2}} and {{math|1=Ω(24) = 3+1 = 4}}.
 
{{math|ω(''n'')}} for {{math|''n''}} = 1, 2, 3, ... is 0, 1, 1, 1, 1, 2, 1, 1, 1, ... {{OEIS|id=A001221}}.
 
{{math|Ω(''n'')}} for {{math|''n''}} = 1, 2, 3, ... is 0, 1, 1, 2, 1, 2, 1, 3, 2, ... {{OEIS|id=A001222}}.
 
== See also ==
* [[Composite number]]
* [[Divisor]]
* [[Table of prime factors]]
* [[Sieve of Eratosthenes]]
* [[Erdős–Kac theorem]]
* [[Arithmetic_function#Ω(n), ω(n), νp(n) – prime power decomposition|Ω(n), ω(n), νp(n) – prime power decomposition]]
 
== References ==
{{reflist}}
 
== External links ==
* [http://www.virtuescience.com/prime-factor-calculator.html Prime Factor Calculator at the Database of Number Correlations]
* [http://www.se16.info/js/factor.htm A Javascript Prime Factor Calculator. Can handle numbers up to about 9&times;10<sup>15</sup>]
* [http://www.javascripter.net/math/calculators/primefactorscalculator.htm Fast Prime Factorization Calculator in JavaScript. Can handle numbers up to 10<sup>20</sup>]
* [http://www.alpertron.com.ar/ECM.HTM Java applet: Factorization using the Elliptic Curve Method finding factors with 20+ digits]
* [http://factors.evalwave.com/fHome.html Millions of factors and prime factors on html pages.]
* [http://dvneo.com/trivia/0505_prime_factors/ List of prime factors for numbers from 2 - 5 million and growing.]
* [http://primefactorizationcalculator.org Prime Factorization Calculator. Can handle numbers up to about 9&times;10<sup>15</sup>]
* [http://www.academia.edu/1066276/Prime_Factorisation_A_New_Approach Prime Factorization A New Approach]
 
{{Divisor classes}}
 
[[Category:Prime numbers]]

Revision as of 14:36, 25 February 2014

For your offense, you might have Gunboats which can easily shoot at enemy safeguarding coming from a prolonged periods of time range and Landing Hobbies which you must fill up when you train gadgets for example Rifleman, Heavy, Zooka, Warrior and Reservoirs. To your village defenses, you might have definitely structures like Mortar, Washer Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Drive Launcher to assist a person eradicate enemies.

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