Re: [Phys-l] Prime numbers

["Ken Caviness" <caviness@southern.edu>]

John's answer is good, but I've always found that a good answer can
still benefit from an example or two.  He mentions unique factorization
theorems.  Allow me to flesh out that reference just a little:

Most people are familiar with the concept of prime factorization.  There
are many different ways to factor most numbers (e.g., 12 = 2x6 and 12 =
3x4), but if we keep breaking down the factors into smaller factors
until they are all prime, we get a unique answer (up to a reordering of
the factors).  (For instance, 12 = 2x2x3.)

If you change the definition of prime from "has exactly 2 factors" to
"is only divisible by 1 and itself", the effect is to allow 1 to be
considered prime, and then we would have an infinite number of
"different" prime factorizations for any number:

12 = 2x2x3
12 = 1x2x2x3
12 = 1x1x2x2x3
...

It might be argued that the exclusion of 1 from the definition of
"prime" is a complication, but it would be much more complicated to have
to define the unique "prime factorization" as the unique "factorization
into primes greater than 1".

Interesting sidenote:  I was helping my 7th grade son with an assignment
for prealgebra class last year, and learned for the first time that
there is a standard extension of prime factorization to negative
integers, listing the first factor as (-1).  So -150 =
(-1)x(2)x(3)x(5^2).  I didn't think much about it at the time, but it
strikes me now that this extension is not unique, since we could replace
the (-1) by (-1)^n, where n is any odd positive integer.  Selecting
(-1)^1 is quite arbitrary.  I wonder whether there are any number theory
results that are more compact, useful or elegant because of considering
prime factorizations of negative integers.  Seems a little bit of a
misnomer, unless we consider (-1) to be prime -- since we're listing it
among the "prime" factors!  If so, is there a benefit to including the
arbitrary "prime factorizations" of 0 (0=0) & 1 (1=1).  I doubt it.

Ken

> -----Original Message-----
> From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
> bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
> Sent: Monday, May 21, 2007 1:45 PM
> To: Forum for Physics Educators
> Subject: Re: [Phys-l] Prime numbers
>
> On 05/21/2007 01:33 PM, Kilmer, Skip wrote:
> > We're getting a bit off track, here, but I've always been curious as
to
> why primes are defined in
> > such a way (exactly two factors) as exclude 1.
>
> That's a deep question, but the answer is shallow:  Things were
> defined that way for convenience, pure and simple.
>
> I hereby define "primoid integer" to be any positive integer that
> has no factors other than itself and unity.  I emphasize that 1 is
> primoid.
>
> It turns out that almost all the interesting results (such as the
> unique factorization theorems) are more conveniently expressed in
> terms of primes rather than primoids.
>
> You most certainly could restate things in terms of "primoids greater
> than 1" but you would probably get tired of it.
>
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