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A '''composite number''' is a [[positive integer]] that has at least one positive [[divisor]] other than one or itself. In other words, a composite number is any positive [[integer]] greater than [[1 (number)|one]] that is ''not'' a [[prime number]].<ref>{{harvtxt|Pettofrezzo|1970|pp=23–24}}</ref><ref>{{harvtxt|Long|1972|p=16}}</ref>
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So, if ''n''&nbsp;>&nbsp;0 is an integer and there are integers 1 < ''a'', ''b'' < ''n'' such that ''n'' = ''a'' × ''b'', then ''n'' is composite.  By definition, every integer greater than one is either a [[prime number]] or a composite number. The number one is a [[Unit (ring theory)|unit]];<ref>{{harvtxt|Fraleigh|1976|pp=198,266}}</ref><ref>{{harvtxt|Herstein|1964|p=106}}</ref> it is neither prime nor composite. For example, the integer [[14 (number)|14]] is a composite number because it can be factored as [[2 (number)|2]]&nbsp;&times;&nbsp;[[7 (number)|7]]. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself.
 
The first 105 composite numbers {{OEIS|id=A002808}} are
:4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140.
 
Every composite number can be written as the product of two or more (not necessarily distinct) primes;<ref>{{harvtxt|Long|1972|p=16}}</ref> furthermore, this representation is unique up to the order of the factors. This is called the [[fundamental theorem of arithmetic]].<ref>{{harvtxt|Fraleigh|1976|p=270}}</ref><ref>{{harvtxt|Long|1972|p=44}}</ref><ref>{{harvtxt|McCoy|1968|p=85}}</ref><ref>{{harvtxt|Pettofrezzo|1970|p=53}}</ref>
 
There are several known [[primality test]]s that can determine whether a number is prime or composite, without necessarily revealing the factorization of a composite input.
 
==Types==
 
One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a [[semiprime]] or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a [[sphenic number]]. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors. For the latter
:<math>\mu(n) = (-1)^{2x} = 1\,</math>
 
(where μ is the [[Möbius function]] and ''x'' is half the total of prime factors), while for the former
 
:<math>\mu(n) = (-1)^{2x + 1} = -1.\,</math>
 
However for prime numbers, the function also returns −1 and <math>\mu(1) = 1</math>. For a number ''n'' with one or more repeated prime factors,
 
:<math>\mu(n) = 0</math>.<ref>{{harvtxt|Long|1972|p=159}}</ref>
 
If ''all'' the prime factors of a number are repeated it is called a [[powerful number]]. If ''none'' of its prime factors are repeated, it is called [[Square-free integer|squarefree]]. (All prime numbers and 1 are squarefree.)
 
Another way to classify composite numbers is by counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are <math>\{1, p, p^2\}</math>. A number ''n'' that has more divisors than any ''x'' < ''n'' is a [[highly composite number]] (though the first two such numbers are 1 and 2).
 
==See also==
*[[Canonical representation of a positive integer]]
 
==Notes==
{{reflist}}
 
==References==
* {{ citation | first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
* {{ citation | first1 = I. N. | last1 = Herstein | year = 1964 | isbn = 978-1114541016 | title = Topics In Algebra | publisher = [[Blaisdell Publishing Company]] | location = Waltham }}
* {{ citation | first1 = Calvin T. | last1 = Long | year = 1972 | title = Elementary Introduction to Number Theory | edition = 2nd | publisher = [[D. C. Heath and Company]] | location = Lexington | lccn = 77-171950 }}
* {{ citation | first1 = Neal H. | last1 = McCoy | year = 1968 | title = Introduction To Modern Algebra, Revised Edition | publisher = [[Allyn and Bacon]] | location = Boston | lccn = 68-15225 }}
* {{ citation | first1 = Anthony J. | last1 = Pettofrezzo | first2 = Donald R. | last2 = Byrkit | year = 1970 | title = Elements of Number Theory | publisher = [[Prentice Hall]] | location = Englewood Cliffs | lccn = 77-81766 }}
 
== External links ==
* [http://www.alpertron.com.ar/ECM.HTM Java applet: Factorization using the Elliptic Curve Method to find very large composites]
* [http://naturalnumbers.org/composites.html Lists of composites with prime factorization (first 100, 1,000, 10,000, 100,000, and 1,000,000)]
* [http://www.divisorplot.com/index.html Divisor Plot (patterns found in large composite numbers)]
 
 
{{Divisor classes}}
 
[[Category:Prime numbers| Composite]]
[[Category:Integer sequences]]
[[Category:Arithmetic]]
[[Category:Elementary number theory]]

Revision as of 22:28, 4 February 2014

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