# Prime Number

## Prime Number

A **prime number** (often simply called a "prime" for short) is a natural number, a positive integer where p greater 1 that has no positive integer divisors other than 1 and p itself.

More precisely, it has exactly two distinct natural number divisors.

All composites break down uniquely into a product of prime factors.

For example, the only divisors of 11 are 1 and 11, making 11 a prime number. 11 corresponds to the factorization 11 = 11 * 1 which means it cannot really be factored to prime divisors other than 11 and 1.

On the other hand, the number 32 has divisors 1, 2, 4, 6, 8, 16, and 32, making 32 not a prime number, it is a composite number. 32 corresponds to the factorization 32 = 2 ^{5}.

Prime numbers are therefore numbers that cannot be factored or, more precisely, are numbers n whose divisors are trivial and given by exactly 1 and n.

The first 30 prime numbers are the following:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, ...

When looking at individual numbers, the primes seem to be randomly distributed, but the global distribution of primes follows well-defined laws.

#### Number 1

The number 1 is a special case which is considered neither prime nor composite.

The number 1 requires special treatment in so many cases and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own.

If the primes included 1, unique factorization into a product of primes would fail.#### Number 2

With 1 excluded, the smallest prime is therefore 2. And, the number 2 is the only even prime number.

Since two is the only even prime number, the term odd prime refers to all prime numbers greater than two.

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