Composite number

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 one that is not a prime number.^[1]^[2]

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;^[3]^[4] 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;^[5] furthermore, this representation is unique up to the order of the factors. This is called the fundamental theorem of arithmetic.^[6]^[7]^[8]^[9]

There are several known primality tests that can determine whether a number is prime or composite, without necessarily revealing the factorization of a composite input.

1 Types
2 See also
3 Notes
4 References
5 External links

Types[edit]

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

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

However for prime numbers, the function also returns −1 and $\mu(1) = 1$ . For a number n with one or more repeated prime factors,

$\mu(n) = 0$ .^[10]

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 $\{1, p, p^2\}$ . 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).

Notes[edit]

^ Pettofrezzo (1970, pp. 23–24)
^ Long (1972, p. 16)
^ Fraleigh (1976, pp. 198,266)
^ Herstein (1964, p. 106)
^ Long (1972, p. 16)
^ Fraleigh (1976, p. 270)
^ Long (1972, p. 44)
^ McCoy (1968, p. 85)
^ Pettofrezzo (1970, p. 53)
^ Long (1972, p. 159)

References[edit]

Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016
Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225
Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766

External links[edit]

v t e Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual number

Constrained divisor sums	Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hemiperfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Practical number

With many divisors	Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number

Aliquot sequence-related	Untouchable number Amicable number Sociable number Betrothed number

Other sets	Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number

Contents

Types[edit]

See also[edit]

Notes[edit]

References[edit]

External links[edit]