I could be wrong but I believe that a prime number by defintion has to be a natrual number. So two would be the only even prime number. 2i and -2i would not be prime numbers because they not natural numbers.
A lot of it depends on context–the context of whatever ring you are working in (a ring is basically a set of “numbers” along with operations of “addition” and “multiplication” satisfying certain properties).
Given some ring, a prime is a nonzero, nonunit (i.e., not a divisor of 1=the multiplicative identity) p such that whenever p divides the product ab, p must either divide a or b.
If your ring is the set of integers (along with the usual addition and multiplication), then this is equivalent to what everyone is used to.
However, if you’re speaking of the rationals or the reals, then there are no primes, since every nonzero element is a unit.
Back to the specific question, yes, in the ring of integers -2 is a prime (as well as -p for any prime p).
Now about 2i and -2i. Calling such numbers even seems a little unconventional to me (even and odd generally used only to describe integers), but I see what you’re saying. However, I’m not sure that the idea could be extended in any useful way beyond numbers of the form n*i, where n is some integer (i.e. …,-3i, -2i, -i, 0, i, 2i, 3i,…). For example, would 2+3i be even or odd?
Anyway, as to their primeness, again, it depends on the ring in consideration. The natural choice would seem to be the ring of Gaussian integers (numbers of the form a+bi, a and b integers). In that case, no 2i and -2i would not be prime. For example:
I agree with you when you say that in mathematics, everything depends on context. That’s why mathematicians agree upon their definitions before they discourse on a topic. The definition you gave for primes is not at all conventional. Your definition is a property of primes, not their defining characteristic. It is known as Euclid’s Lemma, and follows from this conventional definition of primes: An integer greater than or equal to two is said to be prime, by definition, if its only positive integer divisors are one and itself. There are no negative integer primes, if for no other reason than to introduce negative primes would destroy the uniqueness of factorization in the Fundamental Theorem of Arithmetic. -2 is not a prime.
There are infinitely many rational primes. There are infinitely many real primes. They just happen to all be natural primes.
One may reasonably speak of “primes” in other contexts, such as the irreducable polynomial p(x) in k(x), where k is a field, but some sort of appropriate definition and vocabulary would need to be accepted by mathematicians in the field before such usage became widespread.
The corresponding general concept to ‘only divisors are 1 and itself’ in the natural numbers is ‘only divisors are units and itself’ where units are numbers that divide 1 (eg. iiii=1, -1-1=1). This is called ‘irreducible.’ It is not necessarily the same as prime, but you often try to show that it is, when life gets easier (I hate algebra. Technote: they are the same in an integral domain.)
The problem is that in the natural numbers irreducible and prime are the same, so when the concept of a prime was generalised only one property could inherit the name, and someone presumably flipped a coin, choosing to preserve that primes are what you factorize things into.
Actually, that’s not quite right. For example (from the MathWorld link I posted above), take the ring Z[sqrt(-5)] = {a+b*sqrt(-5):a and b are integers}. 2 is irreducible in this ring, but not prime: 2 divides ( 1 - sqrt(-5) ) ( 1 + sqrt(-5) ) = 6, but does not divide either factor.
Primes and irreducible are the same in unique factorization domains, but not integral domains in general.
Further to Cabbage’s comments on the ring of Gaussin Integers, this link provides the necessary and sufficient conditions for Gaussian Primes. They conform to the normal definitions of primes discussed earlier.
The old definition using the number of factors isn’t really useful in general. The proper definition is that p is prime if and only if p is not invertible and
p | ab => p | a or p | b
That is, given a product of two numbers, if p divides the product it must divide one of the factors. It’s also important to note that the domain is very important. In the integers, the primes are (almost) just what we normally think of them as. In the real numbers, since you can invert any number but zero, there are no primes. In the complex numbers, the same fact holds, so there are no primes.
In the Gaussian integers (complex numbers with integral real and imaginary parts), we may run into a different situation, but it gets a little tricky. 2i does satisfy the condition, but it’s nothing really new since it differs from 2 by a unit (multiplication by an invertible element).
Really what mathematicians talk about are “prime ideals” of rings. A ring is a system within which you can add, subtract, and multiply as usual, but not every element is invertible. The integers are a great example. An ideal is a subset I of the ring such that
[ul]
[li]for any two elements a,b in I, a+b and -a are also in I[/li][li]for any element a in I and r in the whole ring, ra is in I[/li][/ul]
A good example is the subset of integers divisible by some fixed integer. The sum of two is still divisible by that integer and the product with any other integer is divisible as well. No, the prime condition for rings is
Given two ideals A and B and the set AB = {ab | a ? A, b ? B}, if AB ? P then A ? P or B ? P. In addition, P must not be all of R.
Now, defining a prime element to be one whose ideal of multiples is prime, we get back exactly what we had before for integers. 2i and 2 appear to be different, but since they both generate the same prime ideal in the Gaussian integers, we consider them to be the same thing.
*Cabbage, I must apologize. I see that not only is your definition of a prime conventional, it’s actually given in one of my old undergraduate number theory texts. When one is dealing strictly with the ring of integers, the two definitions are logically equivalent. But in more general contexts, the property of Euclid’s Lemma is used to define primes, as you stated before I incorrectly contradicted you.
Damn.
I stand, somewhat shakily, by my other comments, though.
For the non-mathematicians out there, the notions of evenness and oddness are generalized as congruences. We say that integers a and b are congruent mod c (that is, with respect to an integer c) if (a - b)/c is an integer. For any integer n, either n/2 or (n - 1)/2 is an integer, so there are only two types of integers mod 2. One of n/3, (n - 1)/3, and (n - 2)/3 must be an integer, so there are three types of integers mod 3.
In theory, you could extend that definition above to rationals, reals, or even complex numbers. I’ve wondered about it, but haven’t had a chance to play with it. The first problem you run into is that a complex number c can be congruent to a whole hell of a lot (i.e., an infinite number) of things mod 2, not just 0 or 1. So while you can partition C according to division by 2, you don’t end up with anything intuitive.
McGraw Hill forced it on me when I was working with them on a project in middle school math.
Ask McGraw Hill. While you’re asking, ask them why it’s so damned important for a sixth grader to be able distinguish a multiplier from a multiplicand.
Thanks for all the other great thoughts on this… much here that I’ll want to spend some quality time with. Always nice to discover a new layer of complexity below the stuff you thought you understood perfectly well…
The problem is that any nonzero element of any field generates the entire field. C/2C is trivial. What you can do, however, is specify a lattice.
The easy example is in R: pick a real number r and consider two numbers equivalent if they differ by an integral multiple of r (rather than any real multiple). This corresponds to different homomorphisms of Z into Ras Abelian groups. In all cases the result is the 1-torus, though with varying geometries. This turns out to be a very useful object in all sorts of areas.
The next easiest is to pick two linearly independant vectors in R[sup]2[/sup] and consider two vectors equivalent if they differ by a vector in the lattice these two vectors generate. This always gives a 2-torus, though again with varying geometry. If we identify R[sup]2[/sup] with C, we get a complex structure on the torus which, again, varies with the lattice.
From here the concept of discrete group actions on various kinds of spaces keeps generalizing and spawns the huge fields of orbifolds, arithmetic geometry, and arithmetic groups.
Primality in the integers would have to have a new definition, because some unusual properties would show up. For example, take 17 – clearly a natural prime. But its divisors include 1 and 17 from the natural numbers, and also -1 and -17 from the negative integers. And of course nobody can make a useful statement as to whether zero could be considered a factor of any number.
But the definition usually quoted, that primes have no natural factors save themselves and 1, is invalid, as it makes 1 itself a prime, which for good and sufficient reasons in number theory is not acceptable.
Rather, a useful definition is that a prime has exactly two discrete natural factors, i.e., itself and one. One itself then falls into a separate category, being the only number with only one discrete natural factor.
The perfect squares of primes have three discrete natural factors; typical composite numbers have four or more discrete natural factors (mostly an even number of discrete natural factors), as do the cubes of primes; etc. I suspect this system of categorizing numbers by the number of discrete natural factors they have may have some useful implications in number theory.
Well, it does have some implications, but as has been covered, primes are not really considered as characterized by their factorizations except incidentally.
More specifically, the integers form a “Unique Factorization Domain”. That is: any integer can be written uniquely as
n = up[sub]1[/sub][sup]e[sub]1[/sub][/sup]…*p[sub]k[/sub][sup]e[sub]k[/sub][/sup]
where u is a unit (an invertible element), k is uniquely defined, the p[sub]i[/sub] are primes (they generate prime ideals in Z), the e[sub]i[/sub] corresponding to a prime p[sub]i[/sub] is uniquely defined, and the p[sub]i[/sub] appearing are defined uniquely up to rearrangement.
The thing is, there’s not that much room for generalizations since almost all commutative rings fail to be UFDs. Number theory as such tends to focus on primes, since prime ideals have an amazing number of useful properties.