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Why Aren't There Negative Prime Numbers?

Date: 12/10/1999 at 00:12:32
From: Edgar Lopez
Subject: 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.

Date: 12/10/1999 at 19:24:14
From: Doctor Ian
Subject: Re: Prime Numbers

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.)

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?")

Mathematics turns out to have so many practical uses that it's easy to 
forget that the mathematicians who are creating it, in many cases, 
really don't care whether the patterns they are finding have any more 
uses than a song, or a painting.

I hope this helps. Thanks for asking an interesting question. Be sure 
to write back if you have others. 

- Doctor Ian, The Math Forum
Associated Topics:
High School Negative Numbers
High School Number Theory
Middle School Negative Numbers
Middle School Prime Numbers

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