section_path: Definition and examples
section_name: Definition and examples
paragraph_text: Definition and examples
  
  A natural number (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called composite numbers. In other words, n {\displaystyle n} is prime if n {\displaystyle n} items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n {\displaystyle n} dots into a rectangular grid that is more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite.
  
  The divisors of a natural number n {\displaystyle n} are the natural numbers that divide n {\displaystyle n} evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This leads to an equivalent definition of prime numbers: they are the numbers with exactly two positive divisors. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition. Yet another way to express the same thing is that a number n {\displaystyle n} is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 {\displaystyle 2,3,\dots,n-1} divides n {\displaystyle n} evenly.
  
  The first 25 prime numbers (all the prime numbers less than 100) are:
  
  No even number n {\displaystyle n} greater than 2 is prime because any such number can be expressed as the product 2 × n / 2 {\displaystyle 2\times n/2}. Therefore, every prime number other than 2 is an odd number, and is called an odd prime. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5.
  
  The set of all primes is sometimes denoted by P {\displaystyle \mathbf {P} } (a boldface capital P) or by P {\displaystyle \mathbb {P} } (a blackboard bold capital P).