|New record prime (GIMPS): 282,589,933-1 with 24,862,048 digits by P. Laroche, G. Woltman, A. Blosser, et al. (7 Dec 2018).|
Answer One: No.
By the usual definition of prime for integers, negative integers can not be prime.
By this definition, primes are integers greater than one with no positive divisors besides one and itself. Negative numbers are excluded. In fact, they are given no thought.
Answer Two: Yes.
Now suppose we want to bring in the negative numbers: then -a divides b when every a does, so we treat them as essentially the same divisor. This happens because -1 divides 1, which in turn divides everything.
Numbers that divide one are called units. Two numbers a and b for which a is a unit times b are called associates. So the divisors a and -a of b above are associates.
In the same way, -3 and 3 are associates, and in a sense represent the same prime.
So yes, negative integers can be prime (when viewed this way). In fact the integer -p is prime whenever p, but since they are associates, we really do not have any new primes. Let's illustrate this with another example.
The Gaussian integers are the complex numbers a+bi where a and b are both integers. (Here i is the square root of -1). There are four units (integers that divide one) in this number system: 1, -1, i, and -i. So each prime has four associates.
It is possible to create a system in which each primes has infinitely many associates.
Answer Three: It doesn't matter
In more general number fields the confusion above disappears. That is because most of these fields are not principal ideal domains and primes then are represented by ideals, not individual elements. Looked at this way (-3), the set of all multiples of -3, is the same ideal as (3), the set of multiples of 3.
-3 and 3 then generate exactly the same prime ideal.
Another prime page by Chris K. Caldwell <email@example.com>