A composite number is a positive integer that has at least one positive divisor other than
one or the number 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 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 primes; furthermore,
this representation is unique up to the order of the factors. This is called the fundamental
theorem of arithmetic. 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.
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. 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
(where μ is the Möbius function and x is half the total of prime factors), while for
the former
However for prime numbers, the function also returns −1 and . For a number n with one
or more repeated prime factors, .
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.
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 . A number n that has more divisors than any x < n is a highly composite number.
See also Canonical representation of a positive integer
Notes
References Fraleigh, John B., A First Course In Abstract
Algebra, Reading: Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N., Topics In Algebra, Waltham:
Blaisdell Publishing Company, ISBN 978-1114541016 Long, Calvin T., Elementary Introduction to
Number Theory, Lexington: D. C. Heath and Company, LCCN 77-171950
McCoy, Neal H., Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon,
LCCN 68-15225 Pettofrezzo, Anthony J.; Byrkit, Donald R.,
Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766
External links Java applet: Factorization using the Elliptic
Curve Method to find very large composites Lists of composites with prime factorization
Divisor Plot