Composite number

A composite number is a positive integer which has a positive divisor other than one or itself. By definition, every integer greater than one is either a prime number or a composite number. The numbers zero and one are considered to be neither prime nor composite. For example, the integer 14 is a composite number because it can be factored as 2 × 7.

Properties

• All even numbers greater than 2 are composite.
• The smallest composite number is 4.
• Every composite number can be written as the product of (not necessarily distinct) primes. (Fundamental theorem of arithmetic)
• Also, $(n-1)!\,\,\,\equiv \,\,0{\pmod {n}}$ for all composite numbers n > 5. (Wilson's theorem)

Kinds 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 $\mu (1)=1$. For a number n with one or more repeated prime factors, $\mu (n)=0$.

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).

See also

• Perfect number