A prime number is a natural number with exactly two factors: and itself.
Classically, was also counted as a prime number, but that is no longer done; it is too prime to be prime.
A number is prime if and only if it generates a maximal ideal in the rig of natural numbers.
Prime numbers do not quite match the prime elements of , since generates a prime ideal but not a maximal ideal; instead they match the irreducible element?s (Wikipedia).