Well?
If we defined prime numbers to include negative numbers, then we’d lose some important properties of primes, e.g., the unique factorisation of natural numbers into primes. If -3 were a prime, then 9 could be factorised both as 3 x 3 and as -3 x -3.
The answer to this question is yes, if we take it to mean: “Are there negative numbers that are prime elements of the ring of integers (that is, the set of numbers . . . , − 3, − 2, − 1, 0, 1, 2, 3, . . . , combined with the usual operations of addition and multiplication)?”
Why this is the case is linked to the question of why 1 is considered to be neither prime nor composite.
And this is because 1 is a unit. It is a number that can divide all the other integers and still have an integer left over every time. In the world of positive integers, 1 is the only unit. But in the world of all the integers, there are two of them: 1 and − 1.
Units are different from prime and composite numbers, precisely because they can divide all the other numbers. Because of this, numbers that get divided by one or the other of them are, in some sense, the same number.
And we see this with say, 6 and − 6. We can cut them up in a whole bunch of ways:
6 = 2 × 3
6 = − 2 × − 3
− 6 = − 2 × 3
− 6 = 2 × − 3
What all of these have in common is the 2 and the 3. There’s no way to cut up a 6 and end up with a 4 or a 9 or a 5 or anything like that. So, what we’re seeing here is that the negative sign is just kind of there. It’s not really affecting the properties of the numbers in terms of whether they’re prime or not, or what their factors are if they’re composite, like 6 and − 6 are.
Now, with the integers, it’s easy to see the pairs of similar numbers that the two units give rise to, and so, it’s easier to just deal with the positive one, and regard the negative one as essentially a subtraction of the positive one. One of them just looks more of a “canonical” prime. And so, you can say that only positive numbers are prime, because it’s easy enough to account for negative primality. Not that this is required, though, but it can make things easier.
This is harder to do with the complex numbers, which have four units, 1, − 1, i, and − i. With the complex numbers, it’s harder to pick one of them as the “canonical” prime. For example, one set of 4 equivalent primes is 1 + i, − 1 − i, − 1 + i, and 1 − i. So, these sort of questions don’t really arise in the complex numbers.
Sure
First of all, let’s define a unit. A number N is a unit if and only if there exists an m such that Nm = 1.
A prime p is thus defined as a non-unit, non-zero number such that for every factorization p = ab that a or b is a unit.
In the natural & whole numbers, the only unit is 1 which gives us our typical definition of a prime number.
In the rational and real numbers, every non-zero number is a unit therefore there are no primes.
In the integers, the only units are 1 and -1 and so the primes are the primes we customarily think of and their additive inverses so if 3 is prime, so is -3.
In the complex numbers a+bi | a,b are real , every non-zero number is a unit since (a+bi) x ((a-bi)/(a[sup]2[/sup]+b[sup]2[/sup])) = 1. Thus like the rational numbers, there are no primes in the complex numbers.
That’s the Fundamental Theorem of Arithmetic which by necessity is limited to the factorization of natural numbers and would then exclude negatives through the closure property. Like my above post points out, it is very important that you are specific about which number system you are limiting yourself to.
(1+i) is a unit and not prime since (1+i) x (1/2)(1-i) = 1
Now if you were talking about the extension field Z(i) then it is prime but Z(i) is not the same as C.
Z(i) is not a field.
Most commonly, “prime numbers” are discussed in the context of the natural numbers (i.e. the positive integers), where every positive integer greater than 1 is classified as being either prime or composite. This is what “a prime number” means in the most common contexts (to a layman or to a number theorist, as opposed to an algebraist).
But, as TriPolar’s second link, and the “Generalizations” section of the Wikipedia article, and SaintCad’s posts, all note, there are other contexts in which the “prime numbers” of that particular set of numbers are something else.
Whether or not there’s only one way to factor 5, for example, depends on what set of numbers you’re allowed to choose your factors from. 5 = 15 = (-1)(-5) = 4*(1.25) = 10*(0.5) = …
Or 5 = (2 + i)(2 - i) so it is weirdly not prime in the Gaussian integers (a.k.a. Z(i)).
As should be clear by now, it’s just a matter of definitional convention, and thus perhaps not a particularly interesting question. But what I would say is this:
When discussing whether a number is prime or not, the sense of “number” which is usually most important is such that two numbers ought be considered equivalent if both divide each other.
Accordingly: against the background context of the integers, I would happily say that -2, for example, is prime. But I would also say that -2 is the same prime as 2. And similarly, for most purposes where I would want to talk about “primes” and “composites”, I would consider -3 and 3 to be the same prime, -4 and 4 to be the same composite, -5 and 5 to be the same prime, and so on.
But nevermind my last post. Really, for the most substantive discussion, what we should do is this: rather than us all telling you from on high how we happen to use words, instead, you should tell us what you consider the key properties characterizing being a “prime number”, or at least give some indication what it is about the notion of "prime number"s and distinguishing these from other numbers which interests you, and then we may all together consider how the ideas you are interested in interface with consideration of negative numbers.
At first I was wondering if there was any conjecture/postulate/theory that states: “any prime can be constructed by the sum of two squares.”
0+0=0
0+1=1
1+1=2
3?
1+4=5
7?
Not working out too well…unless you use negative squares :dubious:
I then asked myself why a prime itself can’t be negative.
I don’t think so. See here.
First of all, any number (let alone prime) in the form 4n+3 cannot be written as the sum of two square and it is fairly easy to prove.
Second, what do you mean “negative square”? If you mean something like i[sup]2[/sup]=-1 and therefore -1 is a perfect square then you are talking about the complex numbers and there are no primes in the complex.
A positive prime is a sum of two squares if and only if it is 2 or has a remainder of 1 when divided by 4. So 5, 13, 17, 29 are sums of two squares while 3, 7, 11, 19, 23 are not. All positive integers are sums of four squares. And, asymptotically, 5 positive numbers out of every 6 is a sum of three squares.
As said upthread, when you ask whether a negative number is a square, you have to ask your context. Certainly when you come to Gaussian integers (of the form a + bi where a and b are odinary integers and i is the square root of -1), then you really have to allow a and b to be positive and negative. For instance 2 = (1+i)(1-i) is not a Gaussian prime. Nor is 5 = (2+i)(2-i) nor any of the integer primes that are sums of two squares, while all the primes that are not sums of two squares remain prime. And so long as you consider p, -p, ip, -ip to be different representations of the same prime you have unique factorization in the Gaussian integers.
Indistinguishable’s Lemma: Two numbers are equivalent if they divide each other.
Statement:Any two rational non-zero numbers are equivalent.
Let a/b and m/n be any two rational numbers not equal to zero
(a/b) / (m/n) = (an)/(bm) e Q
(m/n) / (a/b) = (am)/(bn) e Q
Quod Erat Demonstratum
There is actually a term for what you are discussing (a number times a unit) and for the life of me I cannot remember the term although IIRC it starts out with ‘co’
Two numbers which divide each other are called “associates” or “associated” [though this is just terminology; we might just as well say “similar” or “equivalent” or whatever, as the linguistic context permits]. And, yes, within a field, all non-zero numbers are associated with each other. So, if all you’ve got to think about is a field, there’s nothing interesting to say about primes: everything is either zero or a unit.
I’ll note that any odd number, and therefore almost all primes, can be written as the difference of two squares.
2k + 1 = k^2 + 2k + 1 - k^2 = (k+1)^2 - k^2