prime number (another Prime Pages' Glossary entries) Glossary: Prime Pages: Top 5000: GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)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. 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: 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, PrimeGapsRelated 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 Chris K. Caldwell © 1999-2020 (all rights reserved)