Composite number


Composite number

A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number.

So, if n > 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 – it is neither prime nor composite. For example, the integer 14 is a composite number because it can be factored as 2 × 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 (sequence A002808 in OEIS) 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; furthermore, this representation is unique up to the order of the factors. This is called the fundamental theorem of arithmetic.

Wilson's theorem provides a test for whether a number is prime or composite:


(n-1)!  \equiv 
\begin{cases}
-1 \pmod{n} \mbox{ if } n \mbox{ is prime}\\
\;\;\; 2 \pmod{n} \mbox{ if } n =4\\
 \;\;\;0 \pmod{n} \mbox{ if } n >4 \mbox{ is composite}.
\end{cases}

Types of composite numbers

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

\mu(n) = (-1)^{2x} = 1\,

(where μ is the Möbius function and x is half the total of prime factors), while for the former

\mu(n) = (-1)^{2x + 1} = -1.\,

Note however that for prime numbers the function also returns -1, and that μ(1) = 1. For a number n with one or more repeated prime factors, μ(n) = 0.

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 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 {1,p,p2}. 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

External links


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Look at other dictionaries:

  • Composite number — Composite Com*pos ite (?; 277), a. [L. compositus made up of parts, p. p. of componere. See {Compound}, v. t., and cf. {Compost}.] 1. Made up of distinct parts or elements; compounded; as, a composite language. [1913 Webster] Happiness, like air… …   The Collaborative International Dictionary of English

  • composite number — n. an integer that can be evenly divided by some whole number other than itself or 1: distinguished from PRIME NUMBER …   English World dictionary

  • composite number — noun an integer that is divisible without remainder by at least one positive integer other than itself and one • Hypernyms: ↑number * * * Math. a number that is a multiple of at least two numbers other than itself and 1. [1720 30] * * * composite …   Useful english dictionary

  • composite number — noun A number that is a the product of at least two numbers other than itself and 1. Ant: prime number …   Wiktionary

  • composite number — Math. a number that is a multiple of at least two numbers other than itself and 1. [1720 30] * * * …   Universalium

  • composite number — compos′ite num′ber n. math. a number that is a multiple of at least two numbers other than itself and 1 • Etymology: 1720–30 …   From formal English to slang

  • composite number — /kɒmpəzət ˈnʌmbə/ (say kompuhzuht numbuh) noun Mathematics an integer greater than 1 which is exactly divisible by some integer other than itself and 1 …   Australian English dictionary

  • composite number —   Helu hana ia …   English-Hawaiian dictionary

  • Highly composite number — This article is about numbers having many divisors. For numbers factorized only to powers of 2, 3, 5 and 7 (also named 7 smooth numbers), see Smooth number. A highly composite number (HCN) is a positive integer with more divisors than any… …   Wikipedia

  • Superior highly composite number — In mathematics, a superior highly composite number is a certain kind of natural number. Formally, a natural number n is called superior highly composite iff there is an ε gt; 0 such that for all natural numbers k ≥… …   Wikipedia


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