Composite number

Composite numbers can be arranged into rectangles but prime numbers cannot.

Demonstration, with Cuisenaire rods, of the divisors of the composite number 10

A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the natural numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 and 7; however, the integers 2 and 3 are not because each can only be divided by one and itself.

The first 25 composite numbers (all up to 36) 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. (sequence A002808 in the OEIS)

Every composite number can be written as the product of two or more (not necessarily distinct) primes. For example, the composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 2³ × 3² × 5; furthermore, this representation is unique up to the order of the factors. This fact is called the fundamental theorem of arithmetic.

There are several known primality tests that can determine whether a number is prime or composite, which do not necessarily reveal the factorization of a composite input.

Grimm's conjecture states that, for every set of consecutive composite numbers, there is an equally sized set of prime numbers, and a bijection mapping each composite in the former set to a prime in the latter set that it is divisible by.