Re: prime numbers and African artifact
Michael Jennings (M.J.Jennings@amtp.cam.ac.uk)
13 Jul 1995 20:06:05 GMT
In article <Pine.HPP.3.91.950713005007.12887H100000@weber.ucsd.edu>,
Daniel Kian Mc Kiernan <dmckiern@weber.ucsd.edu> wrote:
>On Tue, 11 Jul 1995, Alistair J. R. Young wrote:
>
>>Rick Hawkins writes:
>>
>>> But only half credit, since it's the wrong answer. 1 is not prime.
>>
>> Correct me if I'm wrong but if a prime number is only divisible by itself
>> and 1, 1 is prime. What else is it divisible by?
>
>I'm familiar with three definitions of "prime number".
>
>[1] A positive integer divisible only by itself and by 1.
>
>[2] Same as [1] except that the number must also be greater than 1.
>
>[3] Same as [1] or [2] except that the number must also be greater
> than 2.
>
>For my part, I don't care for definitions [2] or [3].
>
Why not? Definition two is the only sensible one, as
using this definition it is possible to express any positive integer
other than one as a unique product of prime numbers. This is why prime
numbers are useful  this result isn't called the fundamental theorem of
arithmetic for nothing. In fact, in one way it is best to use as a
definition:
"The prime numbers are the set of positive integers such
that any positive integer other than one can be created as a unique
product of elements (which can be used more than once each) of the set"
This definition explains why we have such a thing as a 'prime
number'. Unfortunately this is a totally nonconstructive definition.
Other equivalent definitions, such as (2) above are more useful in
determining such things as whether a number is prime, and what the
prime numbers actually are, and are therefore more common.
The number 1 is the multiplicative identity, something quite special
and very important, but something entirely different from a prime number.
Michael.

Michael Jennings
Department of Applied Mathematics and Theoretical Physics
The University of Cambridge. mjj12@damtp.cambridge.ac.uk
"Forrest Gump!! Man, I violently *hated* that reactionary piece of subtle
pseudohip drivel... Then again, I don't even like movies. But Jesus  a
movie that really makes the audience wish they were obedient and stupid??
What gives?? It's like something out of the depths of a Stalinist purge."
 Bruce Sterling
