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 a product of two natural numbers that are both smaller than it. The numbers greater than 1 that are not prime are called composite numbers. In other words, is prime if items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange 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.
Demonstration, with Cuisenaire rods
, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly
The divisors of a natural number are the numbers that divide evenly.
Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This idea leads to a different but equivalent definition of the primes: they are the numbers with exactly two positive divisors, 1 and the number itself.
Yet another way to express the same thing is that a number is prime if it is greater than one and if none of the numbers divides evenly.
The first 25 prime numbers (all the prime numbers less than 100) are:
- 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 (sequence A000040 in the OEIS).
No even number greater than 2 is prime because any such number can be expressed as the product . 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 (a boldface capital P) or by (a blackboard bold capital P).