I don't agree to the above conclusion.

Because, the primes -2 and 2 are not two different primes ... -2 and 2 are exactly the same primes. Did you post this, without reading it ?? --

Originally Posted by mayjune View Post
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.
(Relevant portions in red and bold)

So, ... say the number 4 has ONLY ONE unique representation of prime factorization. You may write is as 2^2 or (-2)^2 doesn't matter. They are the same.

It's like having a reference variable in C++.
If a is a reference to b, then you can use a instead of b and that won't matter 'cuz a is an alias (note this word) of b.
So, -p is an alias of the prime p.

I would like to mention something here, that I very much liked :

You have to get a feel for what mathematics is about. It's not about cranking numbers through equations and getting answers, and it's not about balancing your checkbook or figuring out how long it will take Bill and Janet to mow the lawn if they work together, and it's not about building bridges or telephones or sending spaceships to other planets. Those are all _uses_ of math, but mathematics itself is about searching for patterns.
Source : http://mathforum.org/library/drmath/view/55940.html (Original)
###### (Google Cache, 'coz I could not access the original)