We’ve looked at the basic idea of primes, then at where 0 and 1 fit in. But what about negative integers? Can they be prime? If so, how does that affect the definition? And can you factorize a negative number if you don’t have negative primes?
Negative numbers are not primes …
We’ll start with this question from 1999:
Why Aren't There Negative Prime Numbers? Can negative numbers be prime numbers? In my math book it says that a prime number is a whole number greater than 1 that has only two factors, 1 and itself. So why can't, for example, (-5) be prime? Its factors would be -5 and 1. I don't get it. Is it because 5 and (-1) are also factors of (-5)? I say no because (-1) and (-5) would then be factors of 5, and 5 wouldn't be prime. I don't understand and neither does my teacher. We tried to figure it out, and then she told me to ask you.
Edgar’s definition clearly doesn’t allow negative primes – not only because it specifically says so, but also because, for example, \(-5\) would actually have four factors (\(-5,-1,1,5\)), and so would \(5\). (I think Edgar may be confusing the idea of individual factors with factor pairs.)
Later we’ll consider extending the definition to include negatives; but for now, we have to take the definition as given.
Doctor Ian answered:
Hi Edgar, This is an excellent question, but in order to understand the answer, 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. The most interesting patterns are the ones that hold for the largest classes of numbers. So something that is true for all numbers is more interesting than something that is true for just integers, or just prime numbers, or just numbers smaller than 10, or just the number 17. (Note that 'patterns' are sometimes called 'theorems,' and sometimes called 'properties'. For example, the prime number theorem, which says that any composite number can be broken into a product of primes, is one kind of pattern. The commutative property of addition, which says that you can add numbers in any order, is another kind of pattern.)
So the important thing about primes, even before their usefulness, is the universality of facts about them. If a change in definition would result in major exceptions, it would be problematic.
Before anyone figured out the need for negative numbers, mathematicians had already discovered lots of patterns involving prime numbers. So when negative numbers came along, making some of them prime would have caused a lot of patterns (patterns that looked like "For any prime number, blah blah blah") to stop being true. That would have been annoying without serving any real purpose, and so the definition of primes was adjusted to exclude negative numbers.
This, by the way, is why zero and one aren't prime numbers either, although you can make some good arguments for why they should be. If they were considered prime, then a lot of patterns that now look like "For any prime number, blah blah blah" would have to be changed to "For any prime number except one or zero, blah blah blah." That makes the patterns uglier, and harder to remember. ("Oh, wait - does this pattern for prime numbers also apply to zero?")
What patterns would negative primes break? Certainly one would be the Fundamental Theorem of Arithmetic, which says that any number can be written uniquely as a product of primes. This would not be true if \(2, 3, -2, -3\) were all primes, because \(6=2\times3=-2\times-3\), for example. So we’d at least have to change how we state the patterns when all integers are in view.
…or maybe -1 is special …
A question from Tom in 2000 led to a different possibility:
Is -1 Prime? My class discussed prime numbers. The class noted that -1 has exactly two factors (namely 1 and -1) and asked why it wasn't listed as a prime number. I reiterated the definition, but they weren't satisfied. Do you have any further information on this topic?
This takes the question one step further than Edgar’s: If we just drop the restriction to positive numbers, the definition would allow \(-1\) to be prime (but not 1 or other negative numbers).
I answered first, referring back to that last answer, and adding:
I assume that your definition of prime includes the word "positive," as in our FAQ. If the students are asking why the definition doesn't allow for negative primes, then this and other answers we have given should help.
So, there are really two issues: What their definition says, and why the definition should include that restriction. (Our FAQ says essentially what the first answer two weeks ago said.)
Tom replied:
Thanks for writing back, Dr. Peterson. After writing you, students found the FAQ you enclosed and were appeased. The book we are using, Glencoe, defines a prime number as "A number that has exactly two factors, 1 and the number itself." I showed them a couple of other definitions that include natural numbers, counting numbers, or positive integers. They now want to write to the company and suggest a correction. One of the authors teaches in nearby Unionville-Chaddsford, so we may make this a letter-writing lesson too. Thanks for your help.
So their definition was defective, omitting the restriction to natural numbers (or perhaps taking “number” to mean that).
But then Doctor Floor had something to add:
Dear Tom, I read the discussion you had with Dr. Peterson. Indeed it is true that in the usual definition of a prime, the prime is supposed to be a positive integer, or natural number. However, there are people, and not the least, who do consider -1 to be a prime. Professor John Conway of Princeton is one of them. Although I don't suppose it is desirable to have this in high school textbooks, I would like to point you to a newsgroup discussion thread about this subject: -1 as a Prime (geometry-research, Math Forum) http://mathforum.org/kb/message.jspa?messageID=1093170 A lot of this thread is useless, but the first (and later) messages by John Conway do explain why he considers -1 a prime, and why this is useful in his opinion. I hope you appreciate this addition.
We’ve heard from Professor Conway last week, and also here. The discussion, no longer accessible at the old address, appears to be now found here; it is in response to a criticism of Conway’s explanation in his book The Sensual (quadratic) Form, where he says in the preface,
(Don’t ask me why someone would call infinity a prime within this specialized field.)
His first actual use of this convention is on page 95:
(Take a moment to ponder how different this is from the standard usage.)
Here is what he says in the discussion, responding to a complaint about the first quote:
> he gives a list of four example statements that he claims are stated most concisely if -1 is considered to be a prime. his first example is “every nonzero rational number is uniquely a product of powers of prime numbers p“; apparently he’s too stupid to realize that -6 is both 2*3*(-1) and 2*3*(-1)*(-1)*(-1).
I worded that statement more carefully than you’ve read it.
There is indeed a unique way that -6 is a product of powers of distinct primes – namely those powers are -1, 2, and 3. There are only two powers of -1, namely 1 and -1.
…
[L]et me say that I’m very puzzled indeed as to why my calling -1 a prime so enrages you. I did so for a very real reason, not just for fun. Namely, the theory of quadratic forms largely consists of statements – let me call the typical one “Statement(p)” – that are usually only asserted for the positive primes p = 2,3,5,… . Now in my experience, in almost every case when Statement(p) is true for all positive primes, it’s automatically both meaningful and true for p = -1 too, and moreover, that additional assertion is just as useful.
Of course, it’s also the case that -1 does behave differently to the positive primes, but the way this happens is usually that MORE is true when p = -1, not LESS. If LESS were true, then it would indeed be silly to count -1 as a prime, but since it’s almost always MORE, it’s silly not to.
Let me discuss the simplest such statements in this light.
First, the form of the statement of unique factorization that I prefer is:
The multiplicative group of the non-zero rationals is the direct product of the cyclic subgroups generated by “my” primes, namely -1,2,3,5,… .
The additional thing that’s true here is that the subgroup generated by -1 has order 2.
Of course you can still state this with “your” primes, but then have to say “generated by -1 and the primes”, which makes it just a little bit longer.
…
My point is that this a priori silly idea is actually so sensible a posteriori that inside quadratic form theory it’s well worth while changing the accepted convention. But of course you don’t have to – by all means use the longer statements if you prefer them.
This illustrates the fact that we can modify a definition when it makes things easier in a particular field. Definitions are tools of mathematicians, not their masters; and Conway was known as a particularly creative mathematician. He is not proposing that this redefinition be used everywhere.
… or we can extend the definition
A 1997 question from “Muskrat” broadens the context even further:
Prime or Composite? Wouldn't every number be composite? The reason I think this is because the definition of PRIME is that it only has 2 factors...itself and 1... but what about the decimals? Those are numbers, aren't they? I think the definition of factor or prime and composite needs to be defined better, don't you? Because doesn't every number have another number that can go into it? I have another question: in one of your answers you said 0 was a number, but I don't think it is. Isn't zero considered something else? And about factors: every number has a factor, so zero isn't a number. I rest my case.... even 1! see...there is... .5! or .25!
Yes, his definition has to be improved; it is missing some things.
Doctor Rob answered, first clarifying the terms used:
The definitions of prime and composite numbers are fine the way they are. Probably you haven't seen them written out with precision and in detail. I will make an attempt to clarify this for you below. First of all, we shall speak only of the Natural Numbers, that is, the counting numbers. They are all integers, or whole numbers, and they are all positive. They begin 1, 2, 3, 4, ... . A divisor of a natural number N is a natural number D such that N = D*Q for some unique other natural number Q. A prime number in this set is a number with exactly two divisors. Since the number itself and 1 are always divisors, in order to have just two divisors, the number must be bigger than 1, and it must not have any divisors other than 1 and itself.
This is one missing detail in Muskrat’s understanding: 1 does not have 0.5 as a factor (divisor), because divisors must be integers.
The natural number 1 is very special. It is called a unit, and it is the only natural number that has a natural number reciprocal, that is, a natural number I such that 1*I = 1. A composite number is a natural number which is neither a prime number nor a unit.
So the natural numbers split into these three sets: units, primes, and composite numbers.
The big deal about prime numbers is the Fundamental Theorem of Arithmetic. It says that every natural number can be written uniquely as a product of powers of prime numbers. This is a very important fact, as you might be able to tell by its name! That disposes of most of your objections above. 1 is not a prime because it has only one divisor, itself. Zero, negatives, and decimal fractions are neither prime nor composite because they are not natural numbers. They belong to a larger set, either the Integers, the Rational Numbers, or the Real Numbers.
This covers everything we discussed in the last two weeks.
Redefining for integers
Now, we can move beyond that:
The next question is whether we can extend the notion of a prime number to one of these larger sets. In the case of the Integers, this works pretty well, but we have to be careful! Now there are two units, 1 and -1. To every prime number P in the natural numbers there correspond two integers that are "prime" in the integers: P and -P. These now have exactly FOUR integer divisors: 1, -1, P, and -P divide each of the numbers P and -P, and no other integers do. Notice, however, that there are only two of these divisors that are natural numbers. Likewise to every composite number C in the natural numbers there correspond two integers that are composite in the integers: C and -C.
So if we choose to work with integers, we have to change the definitions, so that a prime has exactly four divisors, rather than two (and a unit has exactly two). This way, we can talk about negative prime numbers – because we are now living fully in the world of integers, not just using old definitions in a new context.
Now we have to worry about zero. Zero is a special case, because it has infinitely many divisors, since every integer except zero divides it. Zero is relegated to a new class, neither unit, nor prime, nor composite. The class is called the zero-divisors. Zero is the only zero-divisor in the integers. What happens to the Fundamental Theorem of Arithmetic in this setting? Now it says that every non-zero integer can be written uniquely as a unit times a product of powers of prime natural numbers (or positive prime numbers).
So we’ve also changed the theorem to apply in this bigger world. (We could instead restate this using both positive and negative primes.)
When we try to extend to the Rational Numbers, we are in big trouble: every non-zero rational number is a unit! The same happens in the real numbers. There are no "prime" numbers and no "composite" numbers in those sets, just units and zero-divisors (zero is the only one).
But did you notice that Conway mentioned rational numbers in his version of the FTA? There may be a way to do this, after all. But it, too, will require changes.
Do negative numbers have a prime factorization?
I’ll close with this 2003 question:
Can A Negative Integer Be Factored Into Primes? Can the number -103,845 have the prime factors of 3, 5, 7, 23, and 43? We find this confusing because we have been told a positive number can have prime factors but a negative number can't.
If there are no negative primes, how can you factor a negative number as a product of primes? This is another context issue.
Doctor Tom answered, emphasizing the concept of unit:
Hi Louisa, It just depends on what you consider to be a prime. It turns out that there is a large collection of mathematical structures that can support something like a prime factorization, and the set of integers is just one example. These are usually called "Euclidean domains" in the area of abstract algebra. What's usually done to avoid confusion is to identify the units of the system. Units are factors of 1. For the integers, the units include 1 and -1. Prime numbers in these systems are said to be equivalent if you can obtain one from another by multiplying by a unit, so with that understanding, the primes of the integers are 2 and -2, 3 and -3, 5 and -5, and so on. Factorization is then not unique, but IS unique up to a multiplication by units.
That is, in calling the factorization unique, we ignore any 1’s or -1’s that might be inserted.
If you just consider the positive integers, the only unit is 1, so the prime factorization is totally unique. If you include the negative integers, then you've got uniqueness only up to units. Often people only consider the positives to avoid this confusion.
As we’ve said, the idea of primes originated in a world that didn’t know about negative numbers, so there was no issue. If we choose to expand our world to the integers, we just have to make this little addition.
But we can expand still further, as we also touched on last week:
Just for fun, here's another Euclidean domain you might like to explore. Let i be the (imaginary) square root of -1. The numbers in the system are the so-called "Gaussian integers"--numbers having the form a + bi, where a and b are integers. There is still prime factorization in the Gaussian integers, and interestingly some of the numbers that were prime in the normal integers are no longer so in the Gaussian integers. For example, the number 2, which is written as 2 + 0i, can be factored into (1 + i)(1 - i). Both 1 + i and 1 - i are primes in the Gaussian integers. To see why the idea of units is important, look at the Gaussian integers. In that system, there are four units: 1, -1, i and -i.
This is left for the reader to explore!
Louisa wrote back:
Dear Dr. Tom, Crystal, Darlene, and I would like to thank you for not only answering our question about factoring a negative number but also for returning it so quickly. We had a parent send us a Dr. Math response on negative numbers not being factors of positive numbers. He felt this meant our bonus question asking for the prime factorization of -103,845 meant it could not be factored. We felt our problem was different than what he was answering. That is when we turned to you. We had no idea your response would be so quick and helpful. Thank you. Louisa
The answer they read may have been Are Negatives Factors of Other Numbers?, from a few months earlier, which was about a somewhat different issue; the answer there was that the meaning of “factor” depends on context, and that for a certain question on a standardized exam, negative factors would probably not be in view.
We see now that the answer to the problem is \(-103,845=-1\times3\times5\times7\times23\times43\), where \(-1\) is not a prime, but a unit.