Composite number

Natural numbers from 0 to 100; the composite numbers are highlighted in green

A composite number is a natural number whose prime factorization consists of at least two factors. They are thus the counterpart to the prime numbers . As well as more general primes in rings , such as B. the whole or the Gaussian numbers are examined, composite numbers can also be considered there.

definition

Be and . Then there is an unambiguous product of prime numbers (except for the order of the factors) , so that with prime numbers , whereby individual prime numbers can also appear several times ( with ). is composed if and only if . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n\geq 2}$ ${\displaystyle n=\prod _{i=1}^{k}{p_{i}}}$ ${\displaystyle {p_{1}},\ldots ,{p_{k}}}$ ${\displaystyle p_{i}=p_{j}}$ ${\displaystyle i\neq j}$ ${\displaystyle n}$ ${\displaystyle k>1}$

properties

• With the exception of the numbers zero and one , every natural number is either prime or composite.
• Every positive even number except two is a compound.
• Compound numbers can also be referred to as near-primes of at least the second order .
• The numbers 0 and 1 are neither prime nor composite. While the 0 is completely excluded from these considerations, the empty product is assigned to the 1 .
• The sequence of composite numbers begins with: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25 ... (sequence A002808 in OEIS )

sentences

• Fundamental theorem of arithmetic
• For every composite number :, see also Wilson's theorem ${\displaystyle n>5}$ ${\displaystyle (n-1)!\equiv 0{\pmod {n}}}$

See also

• highly composite number

Web links

• Compound numbers and their divisibility