Compound number
Composite number is any non-prime natural number, with the exception of 1. That is, it has one or more divisors other than 1 and itself. The term divisible is also used to refer to these numbers.
The first seventy-three compound numbers before one hundred 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 9596 98, 99.
Features
One feature is that each can be written as a product of two natural numbers less than itself. Thus, the number 20 is composite because it can be expressed as 4×5; and also 87 since it is expressed as 3×29. However, it is not possible to do the same with 17 or 23 because they are prime numbers. Each composite number can be expressed as a multiplication of two (or more) specific prime numbers, the process of which is known as factoring. The smallest composite number is 4.
The easiest way to prove that a number n is composite, is to find a divisor d between 1 and n (1 < d < n). For example, 219 is composite because it has 3 as a divisor. And also 371 because it has 7 as a divisor. A good alternative is then to use Fermat's little theorem, or rather the generalization of this theorem due to the Swiss mathematician Leonhard Euler.
Since prime and composite numbers are intermingled with each other, it is logical to ask whether there will be sequences of consecutive composite numbers of arbitrary length. The sequence 32, 33, 34, 35, and 36 is an example of length 5, and 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, and 126 an example of length 13. The answer is that we can get as long a sequence of composite numbers as we want. If we want a sequence of length 20, it is enough to take the numbers 21!+2, 21!+3, 21!+4,... 21!+21, since the first is divisible by 2, the second by 3, etc..
A Fermat theorem states that if p is a prime of the form 4n+1, then there is a simple exclusion case, which can be uniquely expressed as the sum of two squares. If a number of the form 4n+1 can be expressed as the sum of two squares in at least two different ways, then the number is composite. Euler found a method of factoring from this fact. For example, if 221 = 112 + 102 = 142 + 52, then 142 - 112 = 102 - 52. Taking gcd(14+11, 10+5) = gcd(25,15) = 5, and then 25/5 = 5 and 15/5 = 3, and finally 52 + 32 = 25 + 9 = 34, so gcd(221, 34) = 17 gives us the factor we are looking for. 1 and 0 are special cases and are not considered prime or composite.