TITLE: Do we have negative prime numbers?
QUESTION [174 upvotes]: Do we have negative prime numbers?

$..., -7, -5, -3, -2, ...$

REPLY [9 votes]: I had come across this page before, but today I'm here because of a duplicate, which I will now answer here.

Generally, the definition of prime numbers is all those natural numbers greater than 1, having only two divisiors [sic], the number itself and 1. But, can the negative integers also be thought of in the same way? 

Almost, but not quite. We could say that a negative integer is prime if it has only two divisors ($-1$ and itself) among the negative integers.
This is very similar to the positive integers, for which we say that a positive integer is prime if it has only two divisors among the positive integers.
For example, 163 is divisible by 1 and itself. It's also divisible by $-163$ and $-1$, but we usually don't bother with that when we are trying to see if 163 is prime.
Similarly to check whether $-163$ is prime, we could try dividing it by $-2, -3, -5, -7, -11$. We don't need to try dividing by $-13$ because $$-13 < (-1) \frac{\sqrt{-163}}{i}.$$
More commonly though, what people usually do is multiply the number by $-1$ and check whether that is prime, and that way they sidestep the issue of imaginary numbers. For example, try Divisors[-163] in Wolfram Alpha.