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alternative definition (prime vs. irreducible)
Beginning with 5*5 = 25, cross out every fifth number, then repeat for the primes 7, 11, 13, etc. until <math>\sqrt{n}</math>.
Beginning with 5*5 = 25, cross out every fifth number, then repeat for the primes 7, 11, 13, etc. until <math>\sqrt{n}</math>.
==Alternative Definition==
Mathematicians prefer the following definition, which is - for integers - equivalent to the one stated above:
<blockquote>''A prime number is a non-unit (i.e., not 1 or -1 for the integers) which whenever it divides the product of two numbers will divide at least one of the factors.''</blockquote>
Or in symbolic notation:
<math>p \in \mathbb{Z} \quad prime :\Leftrightarrow p \not\in \{-1, 1\} \wedge (\forall a,b \in \mathbb{Z}: p \vert ab \Rightarrow p \vert a \vee p \vert b) </math>
The advantage of this formulation is that it can be generalized on other structures which allow for addition and multiplication, i.e., rings.
==Unique Factorization==
==Unique Factorization==
According to the [[fundamental theorem of arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique factorization into prime numbers.
According to the [[fundamental theorem of arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique factorization into prime numbers.