# 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

- 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). - 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.

- Any element which divides one is a
**unit**. - 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). - 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.

**See Also:** PrimeNumberThm, PrimeGaps

**Related pages** (outside of this work)

- Lists of small primes
- Proofs there are infinitely many primes
- Why isn't one prime?
- How many primes are there
less than
*n*? - Home Page for the list of Largest Known Primes
- A Brief Into to Adelic Algebraic Number Theory (especially the section on "factoring primes")
- and of course: The Prime Page home page for info on primes

Printed from the PrimePages <primes.utm.edu> © Chris Caldwell.