# 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

- 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
- and of course: The Prime Page home page for info on primes

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