Can A Negative Integer Be Factored Into Primes?

Date: 11/11/2003 at 16:21:14
From: Louisa
Subject: Can a negative integer be prime factored

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.

Date: 11/11/2003 at 17:23:13
From: Doctor Tom
Subject: Re: Can a negative integer be prime factored

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.

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.

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.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/12/2003 at 10:57:52
From: Louisa
Subject: Thank you (Can a negative integer be prime factored)

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 helfpul.  Thank you.

Louisa