
Glossary: Prime Pages: Top 5000: 
When we seek to develop the integers axiomatically (from a
short list of basic assumptions), we usually include the
WellOrdering Principle as one of these assumptions:
WellOrdering Principle Notice that the positive real numbers do not have this property. For example, there is no smallest positive real number r, because r/2 is a smaller positive real number! The negative integers also lack this property because if r is a negative integer, then r1 is a smaller negative integer. This simple principle of positive integers has many consequences. Let us demonstrate one by proving the following: Theorem: Every integer n greater than one can be written as a product of primes. Proof: Either n is prime (in which case we are done because it is the product of the one prime n), or it has a positive divisor other than one and itself. Let p_{1} be the least of these divisors. Notice that p_{1} must be prime, otherwise there is an integer k with 1 < k < p_{1}, and k divides p_{1}, so k divides n, which contradicts the choice of p_{1}! So n = p_{1}n_{1} where p_{1} is prime and n > n_{1}.This factorization is also unique (up to the order of the factors), see the Fundamental Theorem of Arithmetic.
Chris K. Caldwell © 19992017 (all rights reserved)
