# 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 positive integers 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.