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prime numberAn 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.
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.