Prime Number has 4 Integral Divisors

Theorem

Every prime number $p$ has exactly four integral divisors: $1, -1, p, -p$.

Proof

• From the definition of a prime number, $1$ and $p$ divide $p$.

Also, we have $-1 \backslash p$ and $-p \backslash p$ from One Divides All Integers and Every Integer Divides Its Negative.

• Now suppose $x < 0: x \backslash p$ where $x \ne -1$ and $x \ne -p$.

Then $\left|{x}\right| \backslash x \backslash p$ and $\left|{x}\right|$ is therefore a positive integer other than $1$ and $p$ that divides $p$, which is a contradiction of the conditions of $p$ being a prime.

So $-1$ and $-p$ are the only negative integers that divide $p$, and the result follows.

$\blacksquare$

Notes

Some authors use this as their definition of a prime number.

Thus it needs to be pointed out that by this definition, both positive and negative integers can be prime, i.e. $p$ is prime iff $-p$ is prime.

See Prime Number: Extension to Negative Numbers.