# prime number

An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. For example, the prime divisors of 10 are 2 and 5, and the first six primes are 2, 3, 5, 7, 11, and 13. By the fundamental theorem of arithmetic we know that all integers greater than one factor uniquely into a product of primes.

Technical comment on the definition: In the integers we can easily prove the following

1. A positive integer p, not one, is prime if whenever it divides the product of integers ab, then it divides a or b (perhaps both).
2. A positive integer p, not one, is prime if it can not be decomposed into factors p=ab, neither of which is 1 or -1.
When we study other number systems, these properties may not hold. So in these systems of integers (often called rings) we often make the following definitions:
1. Any element which divides one is a unit.
2. An element p, not a unit, is prime if whenever it divides the product of integers ab, then it divides a or b (perhaps both).
3. An element p, nonzero and not a unit, is called irreducible if it can not be decomposed into factors p=ab, neither of which is a unit.