Equivalence of Definitions of Prime Number

Theorem
The following definitions of prime number are equivalent:

Definition 1 iff Definition 2
This is proved in Prime Number has 4 Integral Divisors:

Definition 1 iff Definition 3
This is proved in Tau of Prime Number:

Definition 1 iff Definition 4
From these two results:
 * $1$ Divides all Integers
 * Integer Divides Itself

it follows that if $p$ has exactly two positive integer divisors then those are $1$ and $p$.

By the same coin, if the only positive integer divisors of $p$ are $1$ and $p$, then $p$ has exactly two positive integer divisors.