A prime number is a natural number with exactly two factors: $1$ and itself.

Classically, $1$ was also counted as a prime number, but that is no longer done; it is too prime to be prime.

Properties

Relation to ideals

A number is prime if and only if it generates a maximal ideal in the rig$\mathbb{N}$ of natural numbers.

Prime numbers do not quite match the prime elements of $\mathbb{N}$, since $0$ generates a prime ideal but not a maximal ideal; instead they match the irreducible element?s (Wikipedia).