Prime

A prime is an integer $p > 1$ whose only positive divisors are $1$ and $p$. Equivalently, $p$ is prime if whenever $p \mid ab$, then $p \mid a$ or $p \mid b$.

The fundamental theorem of arithmetic states that every integer greater than $1$ factors uniquely as a product of primes. This makes primes the multiplicative building blocks of the integers. There are infinitely many primes (Euclid’s theorem).

In ring theory, the concept generalizes: an element $p$ of an integral domain is prime if $p \neq 0$, $p$ is not a unit, and $p \mid ab$ implies $p \mid a$ or $p \mid b$.